MIMO Signals and Systems - PDF Free Download (2024)

Parametric representations of ellipses are of the form p(t) = cosit)yi+sin{t)\2

(2.183)

where vectors Vi and V2 are the major and minor semi-axes. Other parametric equations used to display an ellipse include (see, e.g., [12]) the rotational parametric representation as a two-dimensional vector T?it) = -^(a(l-t^),2btf 1+r^

(2.184) '

or the representation in polar coordinates given by r=

, "^ ^|a^sin^(0) + b^cos^(0)

(2.185)

where range r is expressed as a function of elevation angle 0. If the major axis of an ellipse is inclined, relative to the positive x axis, by an angle of a, the slope of the minor axis is -1/a . In that case, different from Eq. (2.179), the analytic equation for the ellipse rotated about the origin is given by

92

Chapter 2

2

2

^+y-a^

+ ^- = \, a>b,c^^O. b^

(2.186)

&•

MATLAB program "elliptica.m" on the CD ROM can be used to study — for given amplitudes {E^ and E^ and phases (^^ and cpy) of the E vector — the electric field vector changes with elapsing time t or, equivalently, with distance along the z axis. Required inputs are the two amplitudes Ey^ and Ey of the time-harmonic electric field, and the phase angles (py^ and (py in degrees. It is assumed that the field propagates in the positive z direction and that the two orthogonal vector components are phase shifted by ^V = (px-Vy

(2.187)

With a minor modification, the direction of propagation can be changed fi-om +z to -z by simply inverting in the source code the sign of time vector t and time-shifting the origin by the number (F) of full cycles. Another drawing option would be to change the number of full cycles (i.e., P-liC) to be displayed in the 3D diagram on the right. The figure shown on the top left-hand side represents a projection of the E field vector tip onto the (x,y) plane, i.e., at z = 0. If we denote the vertices of the tilted ellipse on the top left-hand side by max\\E^y{ty\ = E^ and max\\E {tVj = Ey, respectively, then the tilt angle 5y. of the ellipse, relative to the positive x axis, can be calculated to be

5^ =—tan ^ 2

x'^y \^X

(2.188)

COS \A(p)

+Ey

The length of the major semi-axis is given by

a = \ME^

+EI+

pUEt

+ 2E^E^yCosi2A

(2.189)

2. Analysis of Space-Time Signals

2\EI+EI-

93

E^+E^ + lEJEt cos{2A

As an illustrative example, let us consider a right-handed elliptically polarized plane wave propagating (clockwise) in the positive z direction. Let the amplitudes along the x and y axes of the tilted ellipse be chosen as E^=\.l and Ey =0.7. Furthermore, let us choose phases (Px=^° and cpy=- 30° , i.e, a phase difference of A(p = (py.-

fcy -I 0.6

^

;---

H-i----:

0.4

£ y 0.2

/^SxY

t ° -0.2 -0.4 -0.6 -0.8 -2

e

Propagation

(a)

t

1.5-:

ib)

7;i;\" T^X---Tr\

1 -05 - -

t^ X/X

1

.\-/-

0 -

Z/X

ic) Figure 2-7. Electric field vector E of right-hand elliptically polarized wave propagating (clockwise) in positive z direction. Parameters; E^ =1-7, E =0.7, Aip = 'iQ°.{a)E.^znAEy components at z = 0 as function of time, tracing out an ellipse tilted by angle S^ relative to positive X axis, (b) Tip of E vector as the wave propagates along the positive z axis; distances measured in multiples of wavelength X . (c) Magnitude changes of E vector along the z axis.

MATLAB program "elliptica.m" also calculates semi-axes a and b, and tilt angle 8^ of the ellipse shown in Figure 2-l(a). For the example under

Chapter 2

94

consideration, we obtain a = 1.8085, b = 0.3289, and a tilt angle of d^ = 20.33°. Because of the right-handed polarization, the sign of axial ratio AR must be negative. Specifically, we have AR = -alb = -5.4986. The sign of AR would be positive had we chosen a left-handed polarization, characterized by a negative time-phase shift of, say, zl^ = -30°. By simply entering different sets of amplitudes and phases of the two linearly polarized orthogonal electric field vectors, {£,., cp^) and {Ey, cpy), the program can serve as a tool to explore various polarization scenarios. If the E lines are permanently parallel to they axis (i.e, E^=0, Ey>0), the wave is characterized as linearly polarized in the y direction. A more general example of a linearly polarized plane wave is shown in Fig. 2-8. Because of the particular choice of E^ = Ey and Acp = 0°, the tip of the wave's electric field vector traces out a straight line inclined by a tilt angle of Sj^ = 45°. Note in 2-8(c) that the magnitude |E| periodically vanishes at odd multiples of /1/4. If we project the E vector of a circularly polarized wave onto the (x, y) plane of a standard cartesian system (again with z being the direction of propagation), the ellipse "degenerates" into a circle. An example of a lefthand circularly polarized wave is shown in Figure 2-9. Circular polarization implies that there are two identical electrical field strengths (E^=Ey), and the magnitude of phase difference Ag) must be an odd-numbered multiple of 90°. If projected onto the (x, y) plane, the end point of a vector representing the instantaneous electric field now traces out a circle. It is easy to verify that the magnitude |E| of a circularly polarized wave's electric field vector doesn't change as a function of distance z, i.e., |E| = \Re{E^^ expijcot - fe)}u^ + Re{E^ exp(Jcot - fc)}u^ = \E^ cos{a> t -kz + (p^)u^

(2.191) + Ey cos{a>t -kz + q)y)u y'"y\

^x~^y

2. Analysis of Space-Time Signals

95

A circularly polarized wave may hence be defined as a superposition of two orthogonal linearly polarized E field components of equal magnitude and a relative phase difference of an odd multiple of nil. The reader is encouraged (see Problem 2.5) to check out the necessary conditions for amplitude and phase sets of right- and left-handed plane waves. It should be clear from the above and from the theory of conic sections that linear and circular polarizations are special cases of elliptical when the axial ratio ± alb is zero or + oo (—> linear polarization), or ± 1 (—> circular polarization), respectively. Note also that, following Eq. (2.178), the principal time-harmonic behavior of magnetic field vector H is almost identical to the one of electric field vector E. In a lossless hom*ogeneous medium, the two vectors (i.e., their time-harmonic components) are in-phase (over time) and spatially orthogonal to each other After calculation of the cross product, it is readily seen that H is nothing else but a "replica" of E, spatially rotated by 90° and magnitude-scaled by the reciprocal of the medium's wave impedance Z. Vectors E and H could be displayed in the form of two twisted corkscrews where E is always perpendicular to H at each point along the axis of propagation. Therefore, loci of the tip of magnetic field vector H are usually not shown as separate curves. Also associated with each wave is a power flux that is characterized by power density vector S (also known as Poynting vector). The instantaneous value of S is defined in the form of vector product S =ExH.

(2.192)

For time-harmonic waves, we use complex-valued field quantities and the vector of average power density given by S = -i;e{ExH*}

(2.193)

where the overbar (...) stands for time-averaging of all vector components, and the star (...*) denotes that complex conjugates are to be considered of possibly complex-valued vector components.

Chapter 2

96

'=yi.5: 1 Ey 0.5'.

£y ii-'

-/-:

I oi -0.5-

-1 -1-5 -

J -2 -1.5 -1 -0,5 0 0 5 1

£x

xV---

1.5' 2

(a)

t

J^ z/X Propagation

W

,;

\/

\L _1: \'r""r V L 2

1.5

• (c)

z/A

Figure 2-8. Electric field vector E of linearly polarized wave propagating along the positive z direction. Parameters: E^ =1-7, Ey =1.7, A(p = 0°. (a) E^ and Ey components at z = 0 as function of time, tracing out a straight line tilted by angle S^ = 45° relative to positive x axis. {b) Tip of E vector as the wave propagates along the positive z axis; distances measured in multiples of wavelength A . (c) Magnitude changes of E vector along the z axis.

2. Analysis of Space-Time Signals

97

2 6 1

- y 0, i\--t

0-1- \ •1

- -

-

5 -2 L- —--. ^ - — -2 -1.5 -1 -0.5

0,5

1

-i:^2

Propagation

(a)

(b)

I 1.75 1.7

1.65

Z/l

{0)

Figure 2-9. Electric field vector E of left-hand circularly polarized wave propagating (counter-clockwise) in positive z direction. Parameters: E-^ = 1.7, Ey = 1.7, A(p = -90°. (a) Ex and Ey components at z = 0 as function of time, tracing out a circle, {b) Tip of E vector as the wave propagates along the positive z axis; distances measured in multiples of wavelength 2.. (c) Magnitude changes of E vector along the z axis.

Note from the above formulas that the Poynting vector S of a circularly polarized wave in a lossless, hom*ogeneous medium has a single nonvanishing component parallel to the direction of propagation, that is, unit vector Uz in our example. By letting E^ =Ey =E and A(p = cPx^Vy = ±n7i:/2,n=l,3,5,..., we write the electrical field vector of a circularly polarized wave in the form of E exp{j'(a> t - kz)) E:

±jEexpij{cot-kz))

' (2.194)

0 Since E has no z component, the magnetic field vector is readily calculated as

Chapter 2

(-dEy/dz\ dE^/dz 0

H=

±Eexp{J(a)t-kz)) - j E exp(j{ci) t - kz))

(2.195)

Because of the necessary amplitude and phase conditions of the E field components involved, the average power density vector S has a single nonvanishing component, pointing in the z direction. It is given by

S = ^Re{Exll*} =

^Re{E^Iiy-EyHl}u^ (2.196)

= —Re{— + ~}Uy =

2

Z

P

Z

Therefore, we observe that the power flow associated with this wave type is directed parallel to the z axis. It is constant over both time and space. Things look much more intricate if two or even a larger number of plane waves with different angular frequencies interact, on the same path, through a lossless, linear and hom*ogeneous medium. The tip of the resulting instantaneous E vector of the time-harmonic field is then observed to move along a very strange-looking orbit while traveling through the chosen coordinate system. MATLAB program "elliptica_mult.m" is a software tool to explore vector pictures of these multiple-wave scenarios. With no loss of generality, the z axis is chosen as the common "railway" for sets of plane waves with variable angular frequencies, polarizations, and possibly opposite directions. The program doesn't set an upper limit to the acceptable number of E vectors. Note, however, that the z axis is scaled with respect to a unit wavelength or an angular frequency of one, respectively. To give an illustrative example. Figure 2-10 shows the superposition of two elliptically polarized waves, both traveling along the positive z direction. The angular frequency (or rotational speed of the vector tip) of one of them is three times higher than that of the other one.

2. Analysis of Space-Time Signals

99

Ey 05

-0,5

Z//1-I

(a)

(b)

-/i-V'V\

'•'

VV/'v/-

\ /

1.5

,

.iJ

2

(C)

Figure 2-10. Electric field vector E of vector sum of two right-hand elliptically polarized waves jointly propagating (clockwise) in positive z direction. Angular frequency of second wave is three times higher than first one. Common parameters of both waves; E^ = 1.7, E = 0.7, A(p = 30°. (a) Superposition of is^ and Ey components at z = 0 as function of time, {b) Tip of E vector of superimposed waves with z axis normalized by wavelength li of first wave, (c) Magnitude changes of resulting E vector along positive z axis.

As shown in Figure 2-11, reversing the direction of propagation of the second wave yields a totally different vector picture. While keeping all other parameters fixed, here, we let the first elliptically polarized wave travel along +z whereas the second one propagates in the negative z direction.

Chapter 2

100

•

EyO.S,

\

'

-

.

^y' 0

-1J-

-0.5 -1 -1.5

-

2

2

. Z//1-I

4

(a)

(b)

--/-

r-'V"iT"

ax

' ;\

,-r LS

—

2

T

A

AL-

z/l-i

Figure 2-11. Electric field vector E of vector sum of two right-hand elliptically polarized waves propagating in opposite directions along z axis. Angular frequency of second wave is three times higher than first one. Common parameters of both waves: iJ^ = 1.7, Ey = 0.7, Atp = 30°. (a) Superposition of i?i and Ey components at z = 0 as function of time, (b) Tip of E vector of superimposed waves with z axis normalized by wavelength ii of first wave, (c) Magnitude changes of resulting E vector along positive z axis.

2.5 CHAPTER 2 PROBLEMS

2.1

Derive the second vector wave equation (2.29) using V xE = -/u

and V xH dt

= J + s— . 8t

2. Analysis

2.2

of Space-Time

Signals

101

The complex propagation constant is defined as y = a + jfi . Show that, according to equations (2.41) and (2.42), the attenuation constant in Np/m is given by a = a)^£/xA—A\ + \ —

rad/m is equal to p = w^enA—A\

2.3

+ \—

+1

Calculate the electric and magnetic field components of a field TE to z in a spherical coordinate system using vector potentials

A=0

2.4

- 1 and the phase constant propagation in

F = ¥'u^='f(coi(0)Ur-sm(g>)u0).

In fi-ee space, we have seen that k = CO^SQUQ . For spherical waves in mode «, calculate and plot (using MATLAB) the cutoff radius r^ where kr = n and

n=\,

2, 3. Draw the equivalent circuit diagrams of the wave impedances TE and TM to radius r.

2.5

Check out the necessary amplitude and phase conditions for right- and lefthanded plane waves. Use MATLAB program "elliptica.m" to verify your results.

2.6

Two elliptically polarized plane EM waves of the type shown in Figure 2-7 propagate in opposite directions along the z axis of a cartesian coordinate system. Suppose both waves have an angular frequency of « = 1. Use MATLAB program "elliptica_mult.m" to characterize the polarization and the space-time function of the loci of the resulting electrical field vector E.

102

Chapter!

REFERENCES [I] J. L. Meriam and L. G. Kraige. Engineering Mechanics — Dynamics. John Wiley & Sons, Inc., New York, N.Y., 4* edition, 1998. [2] C. H. Papas. Theory of Electromagnetic Wave Propagation. Dover Publications, Inc., Mineola, N.Y., 1988. [3] C. A. Balanis. Advanced Engineering Electromagnetics. John Wiley & Sons, Inc., New York,N.Y., 1989. [4] R. F. Harrington. Time-Harmonic Electromagnetic Fields. IEEE Press and John Wiley & Sons, Inc. New York, N.Y., 2001. [5] E. Yamash*ta (Editor). Analysis Methods for Electromagnetic Wave Problems — Volume Two. Artech House, Inc., Norwood, MA., 1996. [6] J. A. Stratton. Electromagnetic Theory. McGraw-Hill, New York, N. Y., 1941. [7] C. R. Scott. Field Theory of Acousto-Optic Signal Processing Devices. Artech House, Inc., Norwood, MA., 1992. [8] Computer Simulation Technology (CST) of America. "Exploring a Three-Dimensional Universe", Microwave Journal, Vol. 44, No. 8, August 2001, pp. 138 - 144. (pdf file available at www.cst.de/company/news/press/MWjournal_Aug_2001.pdf). [9] I. S. Gradstein, I. M. Rhyshik. Tables of Series, Products, and Integrals. Vol. 2, Harri Deutsch, Thun, 1981. [10] M. Abramowitz, I. A. Stegun (eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Vol. 55 of National Bureau of Standards Applied Mathematics Series, U. S. Government Printing Office, Washington, D.C., 1964. [II] M. R. Spiegel and J. M. Liu. Schaum's Mathematical Handbook of Formulas and Tables. McGraw-Hill, New York, N.Y., 2°'' edition, 1998. [12] D. Zwillinger. CRC Standard Mathematical Tables and Formulae. CRC Press, Boca Raton, FL, 31" edition, 2002. [13] H. B. Dwight. Tables of Integrals and Other Mathematical Data. Prentice Hall, Inc., Englewood Cliffs, N. J., 4* edition, 1961. [14] S. A. Schelkunoff Electromagnetic Waves. Van Nostrand, Princeton, N.J., 1943. [15] L. J. Chu. "Physical Limitations of Omnidirectional Antennas," Journal of Applied Physics, Vol. 19, December 1948, pp. 1163 - 1175.

2. Analysis of Space-Time Signals

103

[16] J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and F. J. Corbato. Spheroidal Wave Functions. John Wiley & Sons, Inc., New York, N.Y., 1956. [17] L. B. Felsen and N. Marcuvitz. Radiation and Scattering of Waves. Prentice Hall, Englewood Cliffs, N. J., 1973. (Reprinted by IEEE Press, New York, N.Y., 1996.) [18] D. M. Grimes and C. A. Grimes. "Minimum Q of Electrically Small Antennas: A Critical Review," Microwave and Optical Technology Letters, Vol. 28, No. 3, John Wiley & Sons, February 5, 2001, pp. 172 - 177. [19] J. A. Kong. Electromagnetic Wave Theory. John Wiley & Sons, Inc., New York, N.Y.,

Chapter

3

ANTENNAS AND RADIATION 3.1

INTRODUCTION

After having dealt with various solutions of Maxwell's equations in particular coordinate systems, let us now proceed with the problems generally encountered in the transmission and reception of single or multiple electromagnetic (EM) waves. Our description of these waves propagating through MIMO channels includes an overview of • • " "

radiation and reception of EM waves by simple antennas, different zones and approximations describing fields along the distance from an antenna, radiation patterns of antennas, and propagation of EM waves through linear media.

While the first three phenomena have to do with the originating sources of EM fields, the fourth category is closely linked to the physical characteristics and the dynamical behavior of the transmission medium itself More specifically, our aim is to investigate the radiation patterns of both the transmit and receive antennas. We need explicit formulas describing EM fields distant from a single or multiple known sources. Once these "ideal" fields are sufficiently known, inhom*ogeneities and perturbation effects can be modeled that are attributable to the non-ideal behavior of the channel. As the physical properties of wireless channels usually change over time, we also need this theoretical background to generate appropriate models using statistical terms or, at least, valid empirical guidelines.

105

106

Chapters

3.2

SIMPLE ANTENNAS AND THEIR EM FIELDS

Let us start our tour through the realms of wireless signal transmission by considering the EM field emitted by a single transmit antenna. The first task is to somehow couple electromagnetic energy from a transmission line (wire, coaxial cable, waveguide, etc.) to free space. Vice versa, at the receiver, the last task in our chain involves conversion of a plane EM wave into one that is guided by a transmission line. Note that, in compliance with the reciprocity theorem (see, e.g., [1]), the same rules apply for the EM field properties of transmit and receive antennas. We could, at least hypothetically and for reference purposes, assume an infinitesimally small antenna with an omnidirectional characteristic. Such an isotropic radiator may be conceived as a point source radiating with equal power density in all directions. In geometrical terms, points with equal EM power density are on a sphere with the tiny antenna at its centre. Directional antennas, in contrast, radiate or receive EM waves more effectively in some directions than in others. There is an enormous variety of directional antennas, each one with its own geometry and design parameters. Over the past decades, many excellent books have become available on virtually all aspects of practical antenna design. Recommendations include, e.g., [2] - [5]. Theoretical analyses most frequently refer to the classical books by Harrington [6], Balanis [7], Collin [8], to name just a few. To give a simple example of a, we consider an infinitesimally small electric dipole of incremental length Ai-^0, extending along the z axis from z = -A£/2 t o z = Al/2 . The dipole is surrounded by a hom*ogeneous, lossless, and linear medium. It is excited by a current / flowing in the same direction in each half of the antenna. Thus, the dipole moment is lAi. Time-harmonic fields proportional to exp(Ja)t) are assumed, as usual, and the time dependence of vector potentials and fields will not be explicitly shown below. To get the magnetic vector potential A, we write down the Helmholtz equation in the form of V^A + /t^A = - J 9

7

(3.1) 7

where k =ca £fi = {In / X) , and J is the vector of electric current density (current/area, A/m^), due to the radiating source. Vector A is oriented parallel to the flow direction of current /, i.e., we denote A in cartesian coordinates as A = A^\y^. Since the field is generated by a radiating point source, the magnetic vector potential should be spherically symmetric about the origin. It can hence be expressed as a function of radius r in the form of

3. Antennas and Radiation

A = A,ir)u,

107

= A,{cosi0)n^ -sin(0)u@).

(3.2)

With the exception of the origin, i.e., the location of the point source, we may ignore the electric current density J on the right-hand side of (3.1) and find that the two solutions of y'A,.k'A,=^j~[r'^yk%=0

(3.3)

are proportional to exp(-jkr)/r for an outward-traveling wave and exp(jkr)/r for an inward-traveling wave. To find the proportionality constant, say a, we solve (3.1) for A: -^ 0 and obtain A^ =IAl/{A7ir) which yields a = lAi /{An:). Then, for an outward-traveling wave, the magnetic vector potential is given by A=

exp(-jkr)yi^ = 470"

exp{-jkr)[cos(0)u^

-5m(6>)u@). (3.4)

ATTT

Substitution of A into H = VxA

(3.5)

E = -7|fijM + —V(V-A)1 \ cos )

(3.6)

and

yields a single {azimuth-oriented) magnetic field vector component, i.e.. H = / / ^ u ^ = -—exp{-jkr)\ -:r + j - \sin{0)vL0 \y^ r ' An .IA£sin{0) 2X r

(3.7)

^ ., / , 1 I y jkr ^

and an electric field vector E with two {range- and elevation-oriented) nonvanishing components. Vector E can be calculated, using (3.6) and the mathematical rules of vector differential calculus, as

Chapter 3

108

E = £,.u^ +E0 Ug) lAi

expi-jkr)

J Z

. I

A^'J y \r

-+j

=

cos(6')u^

• asrj

COf^

1

r

(0£r~

A^

sin{0)u0

(3.8)

yj

jZ——exp(-Jkr) lAr \

]kr

{jkrf

where Z = ^^/s is the wave impedance of the medium outside the radiating point source.

3.3 FIELD ZONES AND APPROXIMATIONS Depending on the term kr (or the ratio r/1, respectively), we distinguish between the following three zones: • • •

r/1« 1: near field {static) zone, r/1 « 1; intermediate (induction) zone, r/X » 1: far field (radiation) zone.

Close observation of eqs. (3.7) and (3.8) reveals that the fields have totally different properties in these regions. Near the origin, i.e., in the vicinity of the radiating source, the exponential in (3.8) can be replaced by unity, and the field components are quasi-static. Looking at distances far away from the source, the dominating terms in the field equations decay Hke \lr. All terms proportional to 1/r (or l/r\ respectively) may then be

3. Antennas and Radiation

109

neglected. For sufficiently large values of r/X, the radial component of E vanishes. Hence, the two remaining non-zero components are Eei(r,0) and H0{r,&). They are mutually orthogonal, in (time) phase relative to each other, and transverse to radius vector u^. We may, therefore, describe them as radially propagating TEM mode fields. Mathematically, the so-called far field equations of the Hertzian electric dipole take on the forms of lAe H = H0 0.0 = jk .IA(.sin{0) =y— lA r

exp(-jkr)sin(0)u0 , .,. , exp{-}kr)\X0 (3.9)

r»A,.

E = Eg, U0 = 7 — exp{-jkr)sin{0)\i0 4/ir IA£sin(&) = JZ-exp{-jkr)u0 = Z/Z^u^ 2/1 r Conversely, we may approximate the near field equations by considering in eqs. (3.7) and (3.8) only those terms with the largest decay over range r. The complete set of three non-vanishing field components is obtained as lAe

H = / f0^ u^0

•exp(-jkr)sin{0)u0

47ikr

E = E^u^ + E@ Ug, = -jZ

^expi-jkr)\

iTrkr

3

r« A cosi0)u,

(3.10)

+—-—u^

Equating the magnitudes of the H^'s in the two approximations (3.9) and (3.10), respectively, yields a reasonable limit for the transition from near to far field zones. The two magnetic fields are equally strong (neglecting a 90° phase difference) at the near-to-far field range limit

'Vio/ •

1_A k

2n

(3.11)

In the far field, the average power density (Poynting) vector S of an electric dipole with its two time-harmonic field components can be

Chapter 3

no calculated as 1 * 1 I ^ = -Re{E0H_^}n,=—\E0\

|2

(3.12)

u^.

By integrating S over all area elements dA = r sin{0)d0d©\iy. of a sphere with constant radius r, we obtain the average radiated power as \

JT

In

Prad=-~ \ \ ^^ 0=0 (P=0, 2 / 1

4A^

r

r"- sin(&)d(D d0

J

(3.13)

3

0=0

'-' I A

Considering the complex-valued impedance Z^ = i?^ + jX^ feedpoint, we have

at the antenna

which yields the small electric dipole's radiation resistance in a medium with wave impedance Z = -yjji/s as

(3.15) hi free space, we may substitute the intrinsic impedance of vacuum

Z = r] = riQ= 1-^-^ »120;r« 377 ohms

(3.16)

into (3.15) to obtain the following useful approximation of the radiation resistance:

i?. «80;r2f—1 «79of—I D..

'U

(3.17)

3. Antennas and Radiation

111

Note that Ra is independent of the distance from the source, i.e., it doesn't change as a function of range r. Reactance X^, the imaginary part of Z^, can be calculated by integration of the stored evanescent fields. hi a dual manner, the radiation resistance of a magnetic dipole, i.e., a small circular current loop with radius ro, located at the origin, and with magnetic moment M

(3.18)

•-Kr^I

can be approximated, in free space and using far field components

{H0 ,E0),zs,

R„ ^ 320;r'

i 307.6

(3.19)

kO..

Next, we consider the two most important antenna types shown in Figure 3-1. They are linear wire antennas of lengths Ai = X/2 (half-wavelength dipole antenna) and A£ = A/4 (quarter-wavelength monopole antenna over perfectly conducting surface), respectively. ZA

^X/4

z=X/4 z=0

--^7^

z=0

>y

ZY ^J plane: ' perfect ground (infinite plane conductor)

xtz = -X/4 (a)

(b)

Figure 3-1. Basic elements of simple vertical antennas excited at origin, (a) Half-wavelength dipole. (b) Quarter-wavelength monopole over perfect ground.

Each of them is excited by a current source /Q at z = 0. Along the upper line segment 0 < z < / 1 / 4 , the current distributions of both antenna types are sufficiently well approximated as /(z) = /Q cos(kz) = /Q COS{27I:—) .

(3.20)

A

To determine the EM field, we employ the principle of superposition and

112

Chapter 3

integrate over all infinitesimally small electric dipoles along the wire positions. For the purpose of correct integration, we use primed quantities for the source coordinates and unprimed/zeW coordinates. Therefore, the distance vector zip between a point on the antenna (at position p ' ) and a field point (at position p) is defined as 4p = p - p ' .

(3.21)

As before, we begin by generating an appropriate z-oriented magnetic vector potential 1 ^^'^ I{z') I , A = ^ u ^ = -— 1 -r—i-exp(-7A: zip )dz' u^ 4 ^ - A / 4 |^P| \ -X/A\M K

-YrTexp{-jk\Av\)dz' (cos{0)\y^

(3.22) -sin{0)\iQ)

To show details of the geometrical problem, Figure 3-2 considers the two points of interest during integration. Polar source coordinates p'of an infinitesimally small electric dipole are displayed on the z axis, using primed variables. Polar coordinates of EM field vector p are denoted by unprimed variables.

small dipole source coordinates:

p' = {z',0,0')

distance Z/i^^"'^''"'' ; '^^^^-'^

I

field

vector:

' / p = (r, 0, 0)

>y x^

Figure 3-2. Principle of integration over infinitesimally small dipoles along z-oriented thin wire half-wavelength antenna. Spherical source coordinates primed; field coordinates unprimed.

For an arbitrarily chosen azimuth angle

113

x = r sin{©) cos{0) y = r sin{0) sin(0) i

(3.23)

z = r cos(0) we may express the cartesian coordinates of distance vector zip as Ap = (x- x')u^ + iy-y')Uy+(z~

z')u^ (3.24)

= rsin(0)cos(0)u^

+ r sin(0) sin{0)u y + (rcos(0) - z')}!^

To sum up, at an EM field point p of interest, the contributions of all small dipoles, we need the length (magnitude) of zip which is then to be substituted into the integral of (3.22). It is readily seen that |zlp| = ^r^ sin^(0)(sin^i0)

+ cos^{0)) + {rcos{0) - z'f (3.25)

= ^lr^ +z'^

-lrz'cos{0)

which, sufficiently far away from the antenna (i.e., in the far field where (r / z') » 1 ) , can be approximated as \Ap\«r-z'cosi0).

(3.26)

In the far field, the integral in (3.22) may further be simplified by approximating the denominator of the integrand as |zlp|«r. By closer inspection of the phase term exp{-jk\Ap\)«exp{-j27r—)exp{jkz'cos(0)), A

|r|»|z'|<— 4

(3.27)

we note that a sufficiently accurate far field approximation should at least retain the elevation-dependent exponential. Then, for the current distribution given in (3.20), we end up with a much simpler integral for the magnetic vector potential

Chapter 3

114

A = A^u^ «

expj-jkr)

^/^ \ I{z') expijkz' cos{0))dz' u . ;i/4

exp{-jkr) AK

\IQ cos{kz')exp{jkz' cos{0))dz'

r

(3.28)

-/1/4

•\cos{0)\i^ -^/«(6>)u@) which, after solving for the definite integral on the right (see Problem 3.2), further reduces to .^ . ., >. cos(—cos{@)) . . exp{~jkr) ^2 A = A u ^ =^—^ > 2K kr sir?{0)

(3.29)

Finally, by substituting A into (3.5) and (3.6), and retaining only terms -\lr, the magnetic far field component is obtained as 1

/-^

H = (V X A ) « - — ( r ^ 0 ) u ^ = - — ( - r i;m(0)^2)u
rK

J -- j-^exp(-Jkr) 2nr

cos{—cos{0)) U0 sin(0)

Knowing that H « H^vi^ is azimuth-oriented, the electric far field vector must be elevation-oriented with its magnitude being scaled by the medium's wave impedance Z, i.e. ,K

•hZ^^„r E « ZH0 U0 = j^exp(-jkr) 27ir

,,^,

cos{~cos(0)) ^2 ^-— u^ .

(3.31)

sin{&)

For a time-harmonic field and using the above far field approximations, the half-wavelength dipole's average power density is given, in the form of a range-oriented average Poynting vector S , by

3. Antennas and Radiation

S = -ReiE^UQ

115

xl£pU0} = --Re{E0U0 x —^^u,^} (3.32)

1 ^ irfi7<^os^(izcos(0)) 2 , . ^ \p K,, _i£oJ_f •

l-^ / Q

**r

—

^

T -)

'

Integration over all area elements dA = r sin{0)d0d@\x^ with constant radius r then yields the average radiated power as

of a sphere

'Prad= jl S-fi?A 0,0

Io\Z

,n J 2j. cos{—cos(0))

' 8;r 2r 20=00=0

-^—

^ ^ sin{0)

ATI: 0=0

r^sin(0)d0d0

(3.33)

sin{&)

d0^^-^

An

1. 2188

Note that S and A are vectors, and the dot product yields a scalar quantity Prad, the (time-) average radiated power. As there is no straight-forward solution available for the integral on the right of (3.33), one possible way is to make use of the identity n j^ cos2(~cos(0)) , 1 ~^, d0 = -(r + (9^0

sin{0)

2

2ln(7t)-ln(^)-q(2n)) 2

(3.34)

il.21i where x « 0.57721564... is the Euler's constant and

(3.35) 0

S

is the cosine integral which can be evaluated by means of program "cosint.m" in MATLAB's Symbolic Math Toolbox.

116

Chapters

Using (3.33), the radiation resistance of the half-wavelength dipole in free space (Z « 120;T Q) is obtained as ^a,^/2=—r-—1-2188-73.13Q.

(3.36)

In a similar manner, the physical properties of a quarter-wavelength monopole (geometry shown in Figure 3-1 {b)) can be evaluated. Its field is identical to that of the half-wavelength dipole in the upper half-space (z >0). There is no field, however, in the lower half-space (z < 0) below the infinite, perfectly conducting ground plane. Assuming the same amplitude lo and current distribution in the upper half-space, the monopole's average radiated power is only one half of that of the dipole. Its radiation resistance is, therefore, also reduced by 50 per cent, i.e., ^«,A/4=^^a,A/2-36.56 n .

(3.37)

3.4 RADIATION PATTERNS Depending on their E and H field components, all common antennas have characteristic radiation patterns. These patterns typically indicate the antenna's radiation intensities as a function of azimuth (horizontal plane) and elevation (vertical plane), respectively. One possible way to exhibit the angular characteristics of an antenna's far field is to use the radiation field pattern which is a 3D plot of |E(6>,
(3.38)

The reason why the magnitude of E should be normalized and calculated for a fixed range r is that |E(6', ^ ) | may approach infinity for small values of r. As an example, in Figure 3-3, the characteristic E field of a halfwavelength wire dipole is plotted versus both elevation angle 0 and range r, using the far field approximation of (3.31). Note that this approximation of |E(r,6>)| doesn't make sense for small values of r.

3. Antennas and Radiation

117

For a fixed range r and normalized to the maximum of |E| , we display the RFP in a 3D plot over both azimuth and elevation angles (Figure 2-15 (a)) or, since the thin wire dipole's RFP is independent of azimuth, in the elevation plane as a 2D plot of | E ( 6 ' ) | (Figure 3-4 (b)). Alternatively, we may draw a polar diagram of

IE(6')|

(Figure 3-5).

*!

10'

-B

Figure 3-3. Far field approximation of magnitude of 0-oriented electric field vector. Drawn for half-wavelength dipole (thin wire along z axis, in free space).

118

Chapter 3

RFP

(a) Figure 3-4. Radiation field pattern {RFP) of thin wire dipole in free space, (a) 3D plot of normalized magnitude of electric field IEI over both azimuth and elevation angles 0 and 0, respectively, (b) Elevation plane as a 2D plot of normalized |E((9)| .

Amongst RF engineers, it is more customary to show the antenna's radiation power pattern (RPP), which is the magnitude of the time-average power density vector S (= average Poynting vector) at a fixed range r. As the magnitude of S decreases inversely proportional to the square of r, we pre-multiply S by r^ and define the far field RPP as

RPP(0,0) = r^S(r,0,0)\.

(3.39)

3. Antennas and Radiation

119

Figure 3-5. Polar plot of radiation field pattern (RFP) of half-wavelength wire dipole in free space. Normalized magnitude of electric far field displayed as function of elevation angle 0.

Using again our half-wavelength dipole example, S doesn't change over azimuth. Thus, if normalized to the maximum of S at (9 = 90° and expressed in decibels (superscript dB), we obtain 2 , COS \r r7 -2,2

RPPfl'^(&)=^\Olog,Q

8^ ^

(—COS{0))

sin^O)

Ufz %7t

2„2

,^ cos{0)) cos{— = 20/ogio

sin(0)

A polar plot of (3.40) is displayed in Figure 3-6.

(3.40)

120

Chapter 3

\ :>-^ / / I .'-^-/j

/-iv-

Figure 3-6. Radiation power pattern (RPP) of half-wavelengtli thin wire dipole in free space. Elevation plane view of RPP^'j'^^(0).

Other radiation power patterns of thin wire dipoles with variable length ratios At/X can be evaluated using MATLAB program "rpp.m". Note that this program invokes "logpolar.m" which is also available on the CD ROM attached to the book, "logpolar.m" is an extended version of the standard 2D polar plot program "polar.m" included in MATLAB directory "graph2D". The user is asked to enter the ratio A£/Ji of interest and dBrange, the dynamic range (in dB down the RPP maximum). Al is the total length of the dipole under consideration (not just one arm of it!) and dBrange is ten times the logarithm of the dynamic range to be displayed relative to the pattern maximum at 0 dB. Concentric circles are then automatically drawn and labeled in decibels down the maximum. Assuming a sinusoidal current distribution along the z-oriented dipole of arbitrary length Ai, (3.40) needs to be modified (see, e.g., [6] or [7]) as follows: ( .Ai. . ^ ( .Ae.] cos\ n:{—)cos(0) l-cos\ 7r{—)

J?PPJf>((9) = 20/ogiQ(

^

^

~J-

^-Ai).

(3.41)

sin{0) Another RPP example, now with Ai//I = 2, is shown in Figure 3-7. In case of Ai/A = 2 , we find four zeros and four power maxima (0 dB) distributed along the total angular interval of 0 < 6> < 2;r.

3. Antennas and Radiation

121

Figure 3-7. Radiation power pattern (RPP) of two-wavelengths thin wire dipole A£/ A = 2 in free space. Elevation plane view oi

RPP^f\0).

Even more lobes may occur if the relative dipole length Ai/ X goes up. Counting the pattern's total number of lobes, Ni, as a function of zl^/A , we have 21,

^ - L =0

2{2L + \),

M —-i>0

N,

where L •

At

(3.42)

is an integer number obtained by application of the floor

operator |_...J, i.e., by rounding At/X towards zero. Note that some of the sidelobes may become very narrow and small if the ratio AU X is close to an integer number. However, they exist and are clearly recognizable if dBrange is chosen sufficiently large. To give an example, the fan-shaped radiation pattern shown in Figure 3-8 belongs to an electric dipole with a length ratio oi AU X =1.\.

Chapter 3

122

'.. HP-"-'

Figure 3-8. Radiation power pattern (RPP) of thin wire dipole with length ratio Al/ A. = 7.\ in free space. Elevation plane view of

RPP!jf^\0).

Its weak sidelobes next to, e.g, the strong one at 0 = 90° could be overlooked on a linear scale, but are clearly visible on a plot that goes down to 100 dB. The total number of pattern lobes is readily seen to be N^ = 30, as determined by (3.42). The transmit and receive patterns of passive (no amplifiers), reciprocal (i.e., no non-reciprocal elements or materials such as ferrites) antennas are identical. An antenna's capability to focus its main beam in a given direction is usually described by three closely related parameters called directivity (Z)„), power gain (G„), and efficiency (?/„). To calculate, for a given antenna, these three quantities, we recall from solid geometry that there are An * 12.5664 steradians (sr) in a complete unit sphere [9]. A steradian is, by definition a quantitative aspect attributed to a cone with peak O and extending to infinity. Consider a sphere of radius r centered about point O, as shown in Figure 3-9. The solid (or conical) angle Q is defined as the angle subtended at the center of a sphere by an area A on its surface numerically equal to the square of the radius.

3. Antennas and Radiation

123

Figure 3-9. Definition of solid angle Q.

Thus, we have n =~ .

(3.43)

A unit solid angle (Q = I sr) is obtained if A = r . I t encompasses l/4;7r« 7.9577% of the space surrounding a point. Note that the shape of the area A doesn't matter at all. If wish to calculate the solid angle subtended by an arbitrarily shaped surface S, we express Q in the form of the surface area of a unit sphere covered by the projection of S onto that unit sphere. Using spherical coordinates with differential area dA = r sin(0)d0d0n^ of a surface patch, we may write Qina double integral form as \dA\ n= Ij L_i= \\sin{0)d0d0 0,0 r

(3.44)

0,0

where, for a given surface S, the integration must be performed over all elevation angles 0 and azimuth angles 0 of that particular surface. Integration over a complete sphere (i.e., 0 < C P < 2 ; T , O<0<7r) yields Q = 4K « 12.5664 steradians. The radiation intensity Uiso (in watts per unit solid angle) of an isotropically radiating source (one that radiates equally in all directions) is given by the ratio of the average radiated power over the solid angle of the complete sphere, i.e.,

Uiso—-

(3.45) 4n:

Chapter 3

124

A Honisotropic antenna's maximum directivity is obtained as the ratio of the maximum radiation intensity t/„ai over the radiation intensity of the isotropic source. It is a dimensionless quantity and can be expressed as

D„

U„ U.

An Prad

|/o|^Z-wax{r^|s(6),0)|} In I J \^{0,0ir^ 0=00=0

(3.46)

sin{0)d@d0

where RPPmax is the radiation power pattern taken in the directions (angle &, (p) of its maximum. Of course, directivities D^{@,0) can also be calculated in any direction of interest, that is, as a function of given angles @ and (p, respectively. An antenna's maximum power gain {Ga_max) is defined as An times the ratio of the radiation intensity in the direction of the maximum radiation over the average net input power accepted by the antenna. Mathematically stated, we have the dimensionless quantity An

RPPmaxVQ\ ^ %n^ (3.47)

\lQfz-max{r^^{0,0)\] 2nPin where Pin is the time-average input power at the antenna feed terminals. Note that it is convenient to express both the directivity and the power gain in decibels. Another parameter that is readily derived from Da,max and Ga,max is termed antenna efficiency, rja- It is given by the ratio of the maximum power gain (Ga,max) over the maximum directivity (Da,max) or, equivalently, by the ratio of the average radiated power (Prad) over the average input power (Pin ) at the antenna's feed terminals. We observe that Prad ^Pin and G^ „,Q;C S D^ ^^ . A useful definition of an antenna's total efficiency is thus given by

3. Antennas and Radiation

%=7^

125

= - = — = 7r-7crf.

(0^%.?7r.'7crf^l)

(3.48)

where rja may be split up into a product of two efficiencies. The first one, rjr, is due to reflection losses because of possible mismatch between the antenna (complex impedance: Zu,= Ra '^ jX^ and the connecting cable (transmission line impedance Zi}. Using the previously defined complex reflection factor, r, we have

1

I |2

Za-^L

(3.49)

2a+2^ The second factor, rjcd, that potentially decreases the overall efficiency is due to conductive and dielectric losses in the antenna structure. It should be mentioned that rjcd is usually slightly less than one and, for almost all practical antennas, difficult to calculate. It is, therefore, more convenient to neglect these small losses (lossless antenna assumption: tjcd ~ 1) and to consider a small power margin in the overall link budget. An antenna is in resonance if its reactance (= imaginary part of Za) vanishes, i.e., Xa = 0.

EXAMPLE 3-1: We wish to calculate the total efficiency, r],„ and the maximum power gain, Ga.max, of a losslcss half-wavelength electric dipole antenna in free space. The antenna is connected to a SO-ii transmission line. We start by recalling from (3.40) that the radiation power pattern of the dipole is given by COS {—COS{0))

RPPinm

=

\—

•

(3-50)

sin\0) It has its maximum, RPP„^, at an elevation angle of 0 = 90°. Hence, RPP„^ = max{RPPi/2{0)) = RPPx/2i'^/'^) = 1 • The average radiated power was calculated in (3.33) to be

126

Chapter

Prad=

IJS-dA 0,0 cos(^—cos(&)~)

o 22 STT r

I

I

' ' J ATT QIQ

r^sin{0)d0d0

sin(0)

61=0 * = 0

^ _. sin{0)

^0«-Li! AK

(3.51)

1.2188.

The dipole's maximum directivity is, according to (3.46), obtained as

D„

An: Prad

l/oPz] RPP„ """^ 8;r2

\67r^-\lafzRPP„,

i 1.641,

(3.52)

|/orZ-1.2188-8;r^

which is equivalent to D„ „^^ = 10- log(D„ ^„^)» 2.15 dBi . The pseudo-unit "dBi" refers to the fact that A,,m

2Prad

Z •1.2188«73.13n In

(3.53)

is different from the specified transmission line impedance Z^ = 5 0 n . Therefore, due to a (power) reflection factor r , the total antenna efficiency is given by

%='7r = l - d

=1"

i 0.965.

(3.54)

Rn 1 /T + Zf

Conductive and dielectric losses (index cd) of the antenna are neglected, i.e., we assume T/^j = 1 . Finally, we find the maximum power gain of the half-wavelength dipole connected to a 50-n transmission line to be a,max

^a^a.max

^ '-^o^,

(3.55)

which is equivalent to G^ J^^ « 2 dBi. We note that the mismatch between the radiation resistance of the antenna and the line impedance of the feeding cable causes a small power loss of approximately 0.15 dB. Choice of a 75-Q transmission line would significantly reduce the reflection coefficient and thus yield almost zero power loss. Actually, we have also ignored the reactance (= imaginary part) of the antenna's impedance at the input terminals.

3

3. Antennas and Radiation

127

Although the input reactance of a half-wavelength dipole amounts to j42.5 ohms, it can be reduced to zero by making the antenna length slightly shorter than XI2. For a detailed discussion and analysis of input impedances of dipoles, the reader is referred to Chapter 7 of [7].

The sum of an antenna's maximum power gain (in dBi) and the transmit power (in dBm, i.e., referenced to 1 mW) is usually referred to in link budgets as effective isotropic radiated power, or EIRP. We may thus express EIRP (in dBm) as EIRP = 10/og(G„ „,^) + mogiPin / I mW)

(3.56)

where Pin is the time-average input power at the anterma feed terminals. To give an example, in Europe EIRP is generally limited to 20 dBm (or 100 mW) for wireless local area networks (WLAN's). If the maximum antenna gain is 2 dBi, the input power at the antenna feed terminal should then not exceed 18 dBm (or 63.1 mW). If a lossless antenna is used to receive incoming waves with incident power density 5,„ =|S,„|, we may measure the received average power (= Pr) at the load to which it is connected. Using these two quantities, the ratio

%

~

(3.57)

is defined as the effective aperture (or area) of the antenna. A^g is not to be confused with the physical aperture. It may sometimes become much larger than the physical area. A good example is a thin wire dipole which, in terms of its capability to intercept incoming waves, "appears" much larger than its physical size might suggest. The effective aperture of an antenna with known maximum directivity Da,max and total antenna efficiency t]^ can be shown to be o2

-i2

•"^eff~~. ^a^a,max~~. •'•^

An

^a,max-

(,j.JoJ

An:

Looking at the half-wave dipole of example 3-1, we can easily calculate the antenna's effective aperture as a fimction of wavelength X in the form of ^g^«(0.965-1.641-A^)/(4;r) = 0.126-A^. That is, ^,^ increases with the square of wavelength i, but the aperture decreases with the square of

128

Chapter 3

frequency/ Antenna designers should distinguish carefully between the terms isotropic and omnidirectional. As the word ''isotropic" implies, these (hypothetical!) sources radiate equally in all directions, i.e., over the full ranges of both azimuth and elevation angles. Associated with them are power densities that are uniformly distributed around large spheres centered on the source. The directivity of an isotropic reference antenna is, therefore, independent of both azimuth angles {

=

i

(3.59)

for radiation of wavelength 1. In contrast to an isotropic antenna, an omnidirectional source radiates uniformly in all azimuth directions, but nonuniformly in the elevation directions. Its radiation power pattern is, in polar coordinates, independent of

3. Antennas and Radiation

129

47r/{A&x • ^(92), 47ri\S0/71 f/{A0i

•A02l

iA0],A02

in radians)

iA0^,A02

in degrees)

(3-61)

For planar antenna arrays with a single major lobe, Elliott's approximation [10] suggests a factor of 32,400 (instead of 47r(180/7i)^ = 41,253) in the numerator on the left of (3.61). For practical antennas, Balanis [7] recommends a good approximation of the maximum antenna gain in the form of 30,000

Ga,max = , J ,^ A0yA02

• i^&h ^®2 in degrees).

(3.62)

If the main lobe of the radiation pattern has a symmetrical shape, the half-beamwidth angles are equal and can be calculated for a known Ga,„ax by A@~

173 2 ' =, (zl6> = zl6>i=zl6)2 in degrees).

(3.63)

'\l^a,max

3.5 PROPAGATION OF EM WAVES THROUGH LINEAR MEDIA Let us now consider the general scenario usually encountered in fixed (e.g., line-of-sight) or mobile communications. To introduce a few geometrical details, Figure 3-10 shows the positions of the ith transmitter (subscript T,i) and they'th receiver (subscript RJ) denoted by position vectors PT-,,- and puj , respectively.

130

Chapters

z A

direction of wave'^ , propagation

RJ - -y<

-> y i/f* =0 X

Figure 3-10. Geometry of wave propagation fromrthtransmit antenna (position vector pj-,-) toytli receive antenna (position vector p^^) in a polar coordinate reference system.

Assuming a geometrically fixed transmit/receive (Tx/Rx) pair ij , we may calculate the spatial distance dy between the two antennas as the magnitude of the difference vector di,j=pT,i-PR,j\-

(3-64)

To start with the simplest channel model, we choose wave propagation in free space and ideal hardware components (cables, filters, etc.), each one having zero insertion loss. These losses may be considered later on by linear loss factors or dB's on a logarithmic scale, respectively. Then, for a given carrier frequency/i in Hertz, or wavelength AQ=—,

(speedof light:c = 3xl0^m/s),

(3.65)

/o we wish to calculate the received average power

PRJII

at a point described

(in polar coordinates) by vector p ^ ,• and resulting from radiation power Pr.i emitted by the /th transmit antenna. Index / behind the vertical bar in PR,j\i denotes the source number. Recalling (3.58), we know that each antenna has an effective aperture A^ff and a maximum gain of

Ga,max=f?aDa,max=--2Aff=M—)

A^ff •

(3-66)

The effective isotropic radiated power (EIRP) of the rth transmitter is given

3. Antennas and Radiation

131

by EIRPi='PT,i-GT,i

(3.67)

where Gj-,, is the maximum gain of the rth transmit antenna, as compared to an isotropic antenna with unit gain in all directions. In free space, the average power flux density 5, = S;

(in W/m^) decreases inversely

proportional to the squared distance dij from the transmitter, hi the far-field, on a sphere with radius dij around the rth fransmitter, we have

-,, , Si(dij) =

EIRP

I 1^ PT.i-GTi E; • - =^ -i = - ^

W/m^

(3-68)

where Z (= 1207t ohms) is the wave impedance of free space and Ey is the radiating electric field vector in the far-field region. More precisely, if A£ is the largest physical dimension of the transmit antenna, we require that the distance between the /th transmitter and the y'th receiver exceeds the Fraunhofer distance, i.e., ^ 2(zl^)2 dij»— . /to

(3.69)

Multiplication of the average flux density SiXdjj) by the effective aperture

% . i = ^ ^

(3.70)

of the j'th receive antenna (max. antenna gain GRJ) yields the average received power (in watts) as 'PT.i-Grj PRJ = J- • A^ffj =

= PT,i-GT,rGj,j-(

'PT,i-Gj-rGRj-;\l

^

f

Equation (3.71) is widely known, in a two-antenna form omitting overbars

132

Chapters

and indices / and j , as the Friis formula. It relates, for a given carrier wavelength or frequency, respectively, the transmitted power and the received power when two antennas with Imown gains are located dij meters apart from each other. Notice that the Friis model yields sufficiently accurate results in \h& far-field region and in^ree space only. Moreover, in practice, losses due to hardware imperfections need to be considered by a scalar system loss factor, say Ls > 1, in the denominator of (3.71). It is more convenient to write down the average received power in dBm (i.e., normalized to 1 mW) by taking 10 times the logarithms on both sides of (3.71). Then, we have

P^'^f ^ = 1 0 / o g ( - ^ ) + lOlogiGrj) + lOlogiGj^j) ••'

I

m

W

•'

(3.72) - 20/og(47r) - 20/og( ° ' ''•^') where -20/og(^4;T^»-21.98 dB. Another quantity of practical importance is the free space path loss AP^j which can be expressed as the ratio of the received power to the transmitted power in the form of

APi ,• = iM- = GTi- GR ,• • (

)2

(3.73)

If expressed in decibels, after changing the signs of the logarithms to obtain positive quantities, the free space path loss can be written as APff^ = 20 log(47r) + 20 logi ''•') '•' c -mog{GT_i)-\Olog{GR_j)

(3.74)

Once we know the received power and the (matched!) load resistance Ra at the terminals of the receive antenna, we may calculate the rms voltage Vms available across Ra. The open circuit rms voltage of the unloaded antenna is V^=2pR,j-R,.

(3.75)

Hence, if the antenna is connected to /oat/resistance Ra (i.e. load matched to source resistance), the rms voltage across that load resistance becomes

3. Antennas and Radiation

V. rms

133

^ = ^Pj,j.R^. 2

(3.76)

Another quantity of practical interest is the strength of the electric field (in V/m) at the terminals of thej'th receive antenna. Using equations (3.68) (3.71), the magnitude of the electric field vector may be calculated as

|E.,|,A Z£i^^.A &id5£i v/„

(3.77)

where Z = \207tD. is the intrinsic impedance of the transmission medium, i.e., free space. Let us now go through a typical sequence of Tx/Rx antenna calculations. EXAMPLE 3-2: Two half-wave dipoles with maximum gains of cf^^ = G^''^^ = 2 dB shall be used to transmit a sinusoidal carrier signal of frequency yji = 3 GHz and average transmit signal power Pj- = 10 W over a free space distance ofrf= 5 km. We wish to calculate (a) the path loss in dB, {b) the received power in Watts and dBm, (c) the magnitude of the electric field vector at distance d, and {d) the rms voltage available at a load resistance matched to the receive antenna's impedance. The set of quantities that may be derived from our task description includes: •

wavelength of carrier signal in free space /l = c / / o = (3-10 / 3 ' 1 0 ) m = 10cm.

•

radiation resistance of transmit and receive antenna given by (3.36) as Rr, 3/1 « - ^ - 1 . 2 1 8 8 « 7 3 . 1 3 a . '^a.f./l In

•

Tx/Rx antenna gains are Gj = Gg =10^^'° » 1.585 .

(a) Then, using (3.74), index i = 1 for the transmitter, andj = 2 for the receiver, respectively, we find the free space path loss as

^ / ^ ^ ^ =20/og(4;r) + 20/o^(^^5-^)-10fog(Gri)-10%(G;;2)=*112dB. c

(3.78)

(6) The average received power in Watts is obtained from (3.71) as

I'R.i = Pr,i • Gj, • G^2 • ( ,

which is equivalent to

^^

f « 63.6 pW

(3.79)

134

Chapters

-'•fJ"^ = iOlogi-^^) « - 72 dBn, . '^•^ ImW

(3.80)

(c) Using (3.77), we find the magnitude of the electric field vector at the receive antenna as

Eye 2 = ^ ^ —

»4.36mV/m.

(3.81)

(d) Finally, from (3,76), we find that the rms voltage available at the load resistance R,, (matched to the receive antenna's impedance!) is one half of the open circuit voltage F„. Thus, under the specified ideal conditions, we should expect a receiver input voltage of

^'™,=Y = V^^7^»68.2^V,

(3.82)

• -^

3.6

CHAPTER 3 PROBLEMS

3.1

Using an appropriate magnetic vector potential A, calculate the EM field equations of a small circular loop of constant current / a n d with radius A-Q, located in the xy plane at the origin, and with magnetic moment M = Tvrg'^I. (Note the orientation of A should be that of the current, i.e., A should have a single component in the positive azimuth (0) direction.) Show that, in the far field region, the angular (0) dependences of the TEM (to r) field components are the same as those of the electric dipole, but the polarization orientations of E and H are interchanged. Verify the radiation resistance (R^) formula given in (3.19), based on far field approximations in free space.

3.2

Proof the following identity for the integral on the right of eq. (3.28): A,/4 2 cos{~cos(@y) J/Q cos(kz')exp[jkz' cos{0))dz' = ^ . -X/A k sin\0)

^,. , , . , , , , ,, ^bcos(ax) + asin(ax) Hint: ] cos(ax) exp(ox)ax = exp{bx) r

3. Antennas and Radiation

3.3

135

Two half-wave dipoles with maximum gains of G^''^' = G^''^' = 1.5 dB shall be used to transmit a sinusoidal carrier signal of frequency/) =1.8 GHz and average transmit signal power / j - = 12 W over a free space distance of d = 500 m. Calculate (a) the path loss in dB, (6) the received power in Watts and dBm, (c) the magnitude of the electric field vector at distance d, and {d) the rms voltage available at a load resistance matched to the receive antenna's impedance.

REFERENCES [1] S. A. Schelkunoff and H. T. Friis. Antennas: Theory and Practice. John Wiley & Sons, Inc., New York, N.Y., 1952. [2] J. J. Carr. Practical Antenna Handbook. 4* edition, McGraw-Hill, New York, N.Y., 2001. [3] D. Thiel. Switched Parasitic Antennas for Cellular Communications. Artech House Books, London, U.K., 2002. [4] S. N. Makarow. Antenna and EM Modeling With MATLAB. John Wiley & Sons, Inc., New York, N.Y., 2002. [5]

G. Kumar and K. P. Ray. London, U.K., 2003.

Broadband Microstrip Antennas. Artech House Books,

[6] R. F. Harrington. Time-Harmonic Electromagnetic Fields. IEEE Press and John Wiley & Sons, Inc. New York, N.Y., 2001. [7] C. A. Balanis. Antenna Theory. Analysis and Design. Harper & Row, Publishers, Inc., New York, N.Y., 1982. [8] R. E. Collin. Antennas and Radiowave Propagation. McGraw-Hill, New York, N.Y., 1985. [9] The National Institute of Standards and Technology (NIST). Reference on Constants, Units, and Uncertainty. Online at http://physics.nist.gov/cuu/Units/units.html. [10] R. S. Elliott. "Beamwidth and Directivity of Large Scanning Arrays," The Microwave Journal, January 1964, pp. 74 - 82.

Chapter

4

SIGNAL SPACE CONCEPTS AND ALGORITHMS 4.1 INTRODUCTION Nowadays, vector space representations of signals offer indispensable tools in modem communications. Geometric interpretations of timecontinuous and time-discrete signals are particularly useful in classical signal design and synthesis. Moreover, many applications such as optimum classification and detection of signals, probability-of-error calculations, etc. have greatly benefited from a geometry-based perspective of the technical problem.

4.2 GEOMETRY AND APPLICABLE LAWS From a geometric viewpoint, we may visualize a real- or complex-valued finite-energy signal s{t) as a vector in an orthogonal coordinate system. To get an idea of this fundamental concept, we first recall from the basics of vector analysis that two vectors, say a and b, are orthogonal (or perpendicular, i.e., a l b ) to each other if their dot (or scalar) product is zero. By definition of the dot product, we then have a-b = |a||b|co5'(Za,b) = 0, 0 < Z a , b < ; r

(4.1)

because of the phasor Z a , b , or angle between vectors a and b, is 90°. For our purposes, it would be desirable to have a similar definition of orthogonality for pairs of complex-valued functions, or signals of the type s{t) = Re{s{t)}+jlm{s(t)}, each one with a real and an imaginary part. An appropriate mathematical tool is known as the inner product in a signal vector space. For two ordered pairs of time-continuous complex-valued signals, say 5^(?) and Sj,(t), observed over some finite time interval [^i, ^2], the inner product, will be symbolically denoted in the following by (^...). It is defined in integral form as the complex-valued function

137

138

Chapter 4

^-mitllnit))=]§.mit)s^n(t)dt

(4.2)

h where the asterisk (... ) designates the conjugate of a complex variable. When working with inner products of signals, it is necessary to know and observe a few rules. For any real- or complex-valued scalar weighting factor w, we note the scaling laws w s_^ (0,s_„(0 = w (s_^ (0, s„ (0 ,

(4.3)

(^™(0.>v5„(0) = w {sjt),s„(t)).

(4.4)

By inspection of the definite integral in (4.2), it is readily seen that an interchange of the roles ofSj„(t) and Sj,(t), yields the complex conjugate of the inner product. Therefore, the following property of Hermetian symmetry holds

{l„,itls„{t))=^{l,(t),ijt))*.

(4.5)

Now suppose that signal Sj„{t) is itself a superposition of a finite number of, say P, signals xi{t), X2{t), ..., xp{t). Similarly, s„{t) may consist of a sum of Q signals ;i^i(0, HiiO^ •••> J^Q(0- Calculation of the inner product is then accomplished by observing the distributive laws

(^« it), In (0)=[i^p (0, ^„ (0) = s Up it), .„ (o) \/'=l

I

P=^^

'

(4.6)

= U^{t),s_^{t)) + lx2{t),s_„{t)) + ... + {xp{t),s_^{t)),

(^™(0,^„(0) = L ( O . L , « (4.7)

=\^;„(o,ZiW/+(^^(o,^2^o;+-+u^(o,jg(o

4. Signal Space Concepts and Algorithms

139

Combining the above rules, we may conclude that linear combinations of waveforms can be treated as weighted sums on both sides within in the inner product brackets. Suppose Sj„{t) is a weighted sum of P complex-valued signals with complex weighting factors vv„i, Wj„2, ... Wjnp and Sj,{t) is a weighted sum of Q complex-valued signals with weights vv„i, Wj,2, ..., W„Q. The inner product of these two linear combinations of signals is then calculated as a double sum in the form of

{lm(tlln(t))

= (^y^mp^p(0.

'L^nqlqit)

(4.8)

P

Q

* /

p=lq=\

\

Using our knowledge about the calculation of the inner product of two signal segments, we shall now see how these calculations can be used in the context of signal orthogonality. The energy E^ contained in the m\h signal Sj^it) measured over a finite time interval [t\, tj\ is given by the definite integral

Em = h-n,(OlmiOdt

h

= ]\lm^tf

dt .

(4.9)

h

More precisely, E^ is the amount of energy dissipated in a 1 ohm resistance in At = t2-1\ seconds due to a voltage or current signal with waveform

4.3 ORTHOGONALITY OF SIGNALS Any two finite-length observation signals, sj)) with non-zero energy £„ and Sj,{() with non-zero energy E„, respectively, are said to be ortliogonal over a limited time interval \t\, ^2] if

140

h

Chapter 4

*

t,

[En, = E„, m = n

I

0'

m^n

(4.10)

= E^S(m -n) = EjjS{m - n) where the Kronecker delta function fl, S(m-n) = \

w=«

(4.11)

is used for the sake of brevity. We assume further that there exists a set of pairwise mutually orthogonal waveforms U = {ui{t), MzCt), •••} constituting the unit coordinates in our orthogonal coordinate system called vector signal space. Application of (4.10) to any two of these waveforms (e.g., those with indices m and n, respectively) yields lu^(t)ujt)dt = \^ "'"''-«" 10, m^n

(4.12)

1

A natural extension of the concept of orthogonality is the definition of orthonormal signals. Two signals are orthonormal if they are orthogonal and, in addition, each one of them has unit norm. In analogy to the length of a vector in Euclidean space (say, |a| of vector a), the natural norm of a signal is defined as the square root of the signal's energy. Using (4.9), the natural norm of the wth complex-valued signal Sjn{t) is given by

^it)\ = ^{sJt)JJt)) H

*

= V^m = A\lm(t)lm(t)dt

r?i = MKn

(4.13)

12 ^t

It is customary to write Sj„{t) between two double (instead of single) vertical bars. Normalization of a set of signals is always possible by simply dividing each one of the signals by the square root of its energy. If, for instance, energies E^ and E„ of two orthogonal signals $„{{) and Sj,{t), respectively, are unequal one, we may scale the signals appropriately to make them orthonormal to each other. In terms of their inner product, we

4. Signal Space Concepts and Algorithms

141

write a pair of orthonormalized signals as

(1.(0,1„(0)

[ ^

^ / ^

|o, ^ , „

(4.14)

= (^(TM - n)

where, to distinguish normalized from unnormalized signals, superscript tilde Q stands for the property of having unit norm in the given integration time interval [^1, ^2]One of the reasons why we often perform the final normalization step is that orthonormal signal sets can be used to expand, over the same finite observation interval [^i, t-^, an arbitrarily chosen finite-energy waveform, say x{t), into a convergent series. Expressed as an infinite sum, we may write ^(0 by adding up properly weighted versions of all orthonormal signals of that set. Suppose our signal set includes an infinite number of orthonormal complex-valued signals. Let us call the ^th one of these orthonormal waveforms ?^ (?), and weight it by a properly chosen complex factor w^ . Then, summation over all weighted signals yields a perfect representation of an arbitrarily chosen x{t) in the form of the generalized Fourier series CO

x(0=

Hw^liit).

(4.15)

We call the £th complex weighting factor w^ a generalized Fourier coefficient. Note that these coefficients are meaningful only with respect to a unique and a-priori specified set of orthonormal signals. For a known waveform x{t), the coefficients may be calculated by multiplication of both sides of (4.15) by the complex conjugate s_f^{t) and integration over [^i, ^2]Doing this and observing the above rules for the calculation of inner products yields the (. th generalized Fourier coefficient as

w, = \x{t)s_At)dt.

(4.16)

Note that all these coefficients may be calculated individually and that the complex conjugate superscript can be omitted if the orthonormal signal set consists of real-valued waveforms. Also, depending on the x{t) under consideration and the choice of orthonormal signals, the coefficients may be real or complex scalar quantities. In most practical cases, we have available

142

Chapter 4

only a finite number L of orthonormal signals. Clearly, we should then expect a non-zero error between the exact signal x{t) and its approximation xJJ) given by Xait)=i:w^li{t).

(4.17)

In vector notation (bold face letters!), we may rewrite (4.17) in an Ldimensional signal vector space as the dot product of weight vector (transpose superscript 7) W£ =(W|, W2,..., w^)'^

(4.18)

multiplied by orthonormal signal vector SL(0 = ( ^ I ( 0 . ? 2 ( 0 . - , ^ I ( 0 / .

(4.19)

Thus, the approximated signal xjj) may be expressed as ^a(0 = w i - ? i ( 0 .

(4.20)

The approximation quality can be measured in terms of a suitably defined scalar quantity, preferably a definite integral over the squared error or the square root of that integral. In our previously defined signal vector space, we trace out the exact signal x{t), i.e., the one we wish to approximate, and its approximated version Xjiif). After having observed these two waveforms between the limits of [tx, t^, we consider the complex error signal e{t) = x{t) - ii(0 and calculate its natural norm, or error metric, as ti

Ut)\ = ^ ( x ( 0 - x ^ { t \ x ( 0 - x ^ ( O ) =

\e{t)e {t)dt .

(4.21)

Applications of error metrices are quite frequently found in signal processing and communications where minimization of the error signal energy, i.e., the squared error metric, is a major goal. Let us now go through two illustrative examples to learn more about inner products of complex signals, orthogonality, and the process of orthonormalization.

4. Signal Space Concepts and Algorithms

143

EXAMPLE 4-1: Suppose out of a set S of complex signals we take the first two signals

\t{\-j) £l(0 = -{ 0

-l

-1<;<0

= \^-1 {} - j) 0

Their real and imaginary parts are depicted in Figure 4-1. Two alternative ways of displaying the given signals are shown in Figure 4-2. We may either draw them in the form of 5£)-plots with time t representing the third axis or show the trajectories (paths in vector space) viewed after a projection onto the two-dimensional complex signal plane, also known as signal constellation diagram. Are these two complex signals orthogonal to each other over the finite time interval [-1, 1]? If yes, may we classify them as a pair of orthonormal signals? According to (4.10), to be orthogonal, the two signals' inner product should be zero. It is readily seen that this is the case because

(^,(0,^2(0)=

\[ti}-J)ltii-J)\'dt+\W-J)l-ti\-J)\dt -1

(4-23) = 2 \t'^dt - jt^dt

To be orthonormal, the two signals should have unit energy. The energy values are calculated as

El = (^s_^{t), £;(0) = E2 = (^2(0. ^2(0) = ^*i-

(4-24)

Both signals are, in their original forms, not orthonormal. We may, however, gain orthonormality by amplitude scaling each one of them by the reciprocal of the square root of its energy. As required by (4.14), we obtain a pair of orthonormal signals in the form of

J:i(0 = - 7 = i i ( 0 = - V 3 ^ i ( 0 ,

with their inner product being

l2i0 = -j=i2O^^l2(0

(4.25)

Chapter 4

144

Im{s,(t)]

Re{sM)} • 1

1

;

(c)

(d)

Figure 4-1. Complex signals s\{f) and sjf) for Example 4-1. Plots of real parts (a), (c), and imaginary parts (6), (d).

(?,(0,12(0) = } ^ ^ d t -1 V-^i V^:

= \ 1.1(0/2(0* = 0.

(4.26)

To conclude our example, we may ask the question of whether or not the given signals remain orthogonal to each other if they are observed over a different time window.

4. Signal Space Concepts and Algorithms

145

Im{s_m A

(a)

(b)

/mU2«}A

Re{sM)]

(c)

(d)

Figure 4-2. Trajectories of complex signals ^,(0 and £2(0 for Example 4-1. 3£)-plots of real and imaginary parts as functions of time (a), (c); 2Z)-plots of signal constellations (6), (d).

It is easy to verify that changing either the upper or the lower limit of integration would make the signals asymmetric relative to the origin and yield a non-zero inner product of the two. If, for instance, we choose the integration interval as [-1, 0], the inner product is .,(0,^2(0)= j[«0-i)]tO-j)rA=2Jr'A = - 2 ^ 0 .

(4.27)

The signals would then no longer be orthogonal and, of course, would fail to be orthonormal.

In the next example, we shall examine how finite-duration complex waveforms x(t) can be optimally approximated in a finite-dimensional (say, i-dimensional) vector signal space. There are L orthonormal signals ("unit vectors") available. We shall try to synthesize, as closely as possible, a

Chapter 4

146

replica of x{t) in the form of a weighted sum (linear combination) of these signals. Wanted are the elements, or coefficients, of the optimal weighting vector Wt- Moreover, should there remain a non-zero error signal e{t), we shall calculate its metric and the error energy. EXAMPLE 4-2: We consider, from the previous example, the complex signals ?,(<) and ?2(0 of (4.25) which are amplitude-scaled versions of ^i(<) and S2{t), respectively, originally given in (4.22). As was proven in Example 4-1, Jj(<) and ?2(0 constitute a finitedimensional (£ = 2) set of orthonormal signals. Now, using linear combinations of these two signals, we wish to approximate as closely as possible, over time interval [-1, 1], an arbitrarily chosen complex waveform. Our finite-duration "test signal" is mathematically described as

{-t + j)

i(0 =

-l

else

(4.28)

Its real and imaginary parts are separately drawn in Figure 4-3.

Re{x{t)}

Im{x{t))

rs:^:!EJ^ 1/2

-1/2

(a)

(b)

Figure 4-3. Complex test signal x{t) for Example 4-2. Plot of real part {a) and imaginary part (6), respectively.

4. Signal Space Concepts and Algorithms

147

We start by using (4.16) to calculate the two generalized Fourier coefficients as

Wj = \x{t)'s_^{t)dt

- J [-(-?+m-Stii+mat + \[-{-t+2m-St{\+mt 24

(4.29)

V3(7 + y)

and

Wj = \x{t)'s_2{t)dt -1

= I

[^(-?+m{\-St(\+mat+\i~{-t+27)][-iV3/(i+j)]dt

(4.30)

= |V3(l-7).

Next, by means of (4.17), piecewise approximation (denoted by subscript a) oix{t) yields

2a (0 = Hill (0 + If2l2 W =

1 3 -t-j-t 2 4 1 3 —t+j—t 2 2 0

-1<«<0 0

efce

We notice that the error signal

j ( l +| 0 e(t) = x{t)-x(t)

={7(1-|0 0

-\

else

is purely imaginary since the real part of x(/) is perfectly well (zero error!) approximated by xjf). Finally, let us calculate the minimized energy contained in the error signal. We found an optimal approximation xjf) with a minimum error energy of

148

Chapter 4

E, = {e(t},e_(t))= je_(t)e (t)dt (4.33)

= l(-+-tfdt _l 2

+ H\-^fdt

4

2

= 16

and an error metric of

\\e{t)\\ = ^

=~ ^ .

(4.34)

It is left as an additional exercise (see Problem 4.1) to perform, in a straightforward manner, the above steps of piecewise integration, to sketch the real and imaginary parts of x^(t), and to draw a plot of the error signal e(f). >•<

There are a plethora of known classes of orthogonal signal sets. Perhaps the most prominent representatives are complex exponentials observed over a full signal period T. Suppose we consider, over the time window [0, 7], in an infinitedimensional signal space the set of time-continuous complex exponentials l((t)

= exp(j2Kit/T),

i^0,±l,±2,...

(4.35)

We pick out any two of these signals, say Sj„(t) and Sj,{t) with arbitrary indices m and n, and proof their orthogonality by calculating their inner product as T

{lm(t),S_„it))=ls_Jt)s_lit)dt 0 T

= I exp(j 2nmt/T)

exp{-j lnnt/T)dt

(4-36)

T

m=n

m^n

From there it is easy to conclude that, after normalization, complex exponentials

4. Signal Space Concepts and Algorithms

l({t) = -^expU2Kit/T).

€ = 0,±1,±2,...

149

(4.37)

represent a set of orthonormal signals over the time interval [0, 7]. As an interesting detail, we note that the modulus (or absolute value) of the inner product of two signals is always less or equal to the product of the two signals' norms. Mathematically, we write this in the form of Schwarz's inequality as \{s„fi),s„it))\<\\sjt)\\\\s„{t)\\ (4.38)

Equality exists if and only if signal s„(t) is a weighted version of 5^(0, i.e., Sj,{t) = w sj(t) where w is an arbitrary complex weighting factor. There are cases where we wish to extend a given set of real-valued orthogonal signals into a set of complex-valued orthogonal signals. Knowing that the real parts are orthogonal to each other, we need to find and add imaginary signal parts such that the resulting complex signals are orthogonal, too. To give an example, let us consider two real signals, a{t) and c{t), with zero inner product over time interval {tu ^2]- We write the desired orthogonal signals as complex waveforms s_^{t) = a{t) + jb{t),

s_^{t)

= c{t) + jd{t)

(4.39)

where b{t) and d{t) are the two unknown signals. Then, since we know that a{t), c{t)) = I a{t)c{t)dt = 0,

(4.40)

we can make complex signals s_x{t) and S2{t) orthogonal to each other by simply choosing their imaginary parts as bit) = -ait),

dit) = -cit).

(4.41)

This is readily proofed by calculation of the inner product of the two complex signals. Consideration of (4.40) yields

Chapter 4

150

{h (0, ^2 (O) = I («(0 ~j a{t)\c{t) - j c{t))* dt (4.42)

2\a{t)c{t)dt = Q.

So far, we dealt with orthogonality of continuous-time signals. Practical work with discrete-time signals, however, requires a few modifications to be made to the above formulas and mathematical rules. Firstly, we need to replace definite integrals by finite sums and continuous-time signals s_{t) by their sampled, discrete-time counterparts s_{kTs) or, for short, s(Ji) where k is an integer counting index. Sampling time intervals Ts are usually, but not necessarily, being kept constant over time t. Secondly, in digital signal processing, we may retain and even exploit the concept of vector signal spaces. Samples of waveforms are represented by discrete points in a finitedimensional signal space. Compact notations of multiple discrete-time signals are then possible by writing samples of individual signals in vector form and assembling these vectors as rows or columns, respectively, of signal matrices. Throughout the following text, we shall arrange time samples along the columns of a multiple signal matrices S. Suppose we observe A'^ signals in parallel and have available K samples per signal. Then, a multiple signal matrix is defined as

^,(2)

^2(2)

...

s^(2) :(S],S2,...,S^).

(4.43)

The «th column (n = 1, 2, ..., N) of matrix S includes K sample values taken at discrete times kTs {k = 1, 2, ..., K) and representing the wth signal. We shall call s«=(s„(lXl„(2),...,l„(/^)y,

n = \,2,... N

(4.44)

the Mth discrete-time signal vector or, for short, discrete signal vector. If we measure, over a finite time interval [ki, ^2], ^k = k2~k\ samples of the mth discrete-time complex signal Sj„ and group them consecutively in discrete signal vector S;„, then the energy in $„ is given by the sum

4. Signal Space Concepts and Algorithms

E„=

il„,{k)s^„{k)=

151

I \s_^{kf .

k=k\

(4.45)

k=k^

In analogy to the properties of continuous-time signals, we may calculate the inner product of two discrete signal vectors observed over discrete time interval [k\, k^ as a real-valued quantity. It is given by ( s ^ , s „ ) = is_^{k)s^^{k)

\<m,n

Any two discrete signal vectors, s^ and s„, are orthogonal to each other (denoted by s„, -Ls„ ) if their inner product vanishes. In ordinary Euclidean vector space, we would interpret the inner product as the product of the length of vector s^, the length of vector s„, and the cosine of the angle between these two vectors. Geometrically, the natural norm (symbolically denoted by ||...||) of the mth discrete signal vector s„ may be interpreted as its length. It is given by the square root of the inner product of s^ with itself. In other words, we have

l=# (4.47)

U2

.

\k=k^

Following the above interpretation of signals as vectors in Euclidean space and using direction cosines, the angle (p^.n between discrete signal vectors s^ and s„ is obtained as the inverse cosine given by ^'s„