# Ch6 1

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```					Communication Signal Processing

Chapter 6. Spatial Methods
Soongsil University School of Electronic Engineering Woosik Moon
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Outline
 6.1 Introduction  6.2 Array Model  6.2.1 The Modulation-Transmission-Demodulation Process
 Modulation

 Demodulation

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Introduction
 Spatial spectral estimation problem
 The problem of locating n radiating sources by using an array of m passive sensors  This problem basically consists of determining how the “energy” is distributed over space, with the source positions representing points in space with high concentration of energy  Hence, it can be named a spatial spectral estimation problem
Source 1 Source 2 Source 3 Source n

Sensor 1

Sensor 2 Sensor 3 Sensor m

Figure 6.1 The set-up of the source-location problem.

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Introduction
 The development of the array model
 The emphasis in this chapter will be on developing a model for the output signal of the receiving sensor array  The development of the array model is based on a number of simplifying assumptions - The sources are assumed to be situated in the far field of the array - Assumed that Both the sources and the sensors in the array are in the same plane and that the sources are point emitters - Assumed that the propagation medium is homogeneous and so the waves arriving at the array can be considered to be planar - Under these assumptions, the only parameter that characterizes the source locations is the so-called angle of arrival, or direction of arrival (DOA)

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Array Model
 The output of sensor k can be written as
 Let x(t) denote the value of the signal wave form as measured, at time t  Let τ(t) denote the time needed for the wave to travel from the reference point to sensor k (k=1,2,…,m)  hk  t  is the impulse response of the kth sensor  e  t  is an additive noise
k

yk  t   hk  t   x t  k   ek t 
 Let X(ω) denote the Fourier transform of the signal x(t):
X     x  t  e it dt
 

(6.2.1)

(6.2.2)

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Array Model
 The inverse transform, which expresses x(t) as a linear function of X(ω), is given by
x t   1 2







X   eit d

(6.2.3)

 By using the properties of the Fourier transform, Yk   can be written as

Yk    Hk   X   ei k  Ek  

(6.2.4)

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Array Model
X  
2

S  

2

c

0

c



0



Figure 6.2 The energy spectrum of a bandpass signal.

 The general class of physical signal, such as the carriermodulated signals encountered in communications
     ωc denotes the center frequency Assume that the received signal x(t) is bandpass signal Let s(t) denote the baseband signal associated with x(t) The process of obtaining x(t) from s(t) is called modulation The inverse process is called demodulation

Figure 6.3 The baseband spectrum that gives rise to the bandpass spectrum in Figure 6.2.

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The Modulation-Transmission-Demodulation Process
 The physical signal x(t) is real valued; hence, its spectrum |X(ω)|2 should be even (i.e., symmetric about ω=0)  The spectrum of the demodulated signal s(t) might not be even; hence, s(t) might be complex valued

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The Modulation Process
 The complex modulation process







s  t  eict e  it d    s  t  e




 i  c t

d   S    c 

(6.2.5)

 The real modulation process

X    S   c   S      c 
 The real modulation process in the time domain
x t   1 2 1  S   c   S     c   eit d     2


(6.2.6)







S   c  e 

i  c t ic t

e d

 1   2







S    c  e

 i  c t ic t

  e d   s  t  eict   s  t  eict    



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The Modulation Process
 Which gives

x  t   2Re  s  t  eict   

(6.2.7)

 or

x t   2 t  cos ct   t 

(6.2.8)

 Where   t  and   t  are the amplitude and phase of s  t  , respectively,

s t    t  e

i  t 

 If we let sI  t  and sQ  t  denote the real and imaginary parts of s  t  , then we can also write (6.2.7) as

x  t   2  sI  t  cos ct   sQ  t  sin ct   
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(6.2.9)

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The Demodulation Process
i t  The baseband signal s  t  can then be obtained by filtering x  t  e c with a baseband filter whose bandwidth is matched to that of S  .







x t  e

 ic t  it

e

d   x  t  e




 i  c t

d   X    c 

X   c    S    S     2c    

Figure 6.4 A simplified block diagram of the analog processing in a receiving-array element.
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