The coherence time Tc for the random channel is de ned to be value of Dt such that RH(Dt) < 0.5.
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BASIC CONCEPTS IN WIRELESS COMMUNICATIONS
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RH (t,n) Time frequency autocorrelation
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Fourier transform pairs
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t n=0 RH (t) t=0 RH (n)
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Temporal autocorrelation
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Frequency autocorrelation
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Time t
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Frequency n
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SH (f )
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SH (t)
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Doppler spectrum
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Delay spectrum
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Doppler frequency f
Delay t
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SH (f, t )
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t Delay Doppler spectrum
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Figure 1.5. Time frequency autocorrelation and delay Doppler spectrum.
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Similarly, the correlation in the frequency dimension is given by RH (D ) = RH (Dt , D ) Dt =0 (1.17)
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The coherence time Bc for the random channel is de ned to be value of D such that RH(D ) < 0.5.
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WIRELESS CHANNEL MODELS
On the other hand, we can characterize the random channel on the basis of the delay Doppler spectrum. For instance, the Doppler spectrum is given by SH ( f ) = SH ( f , t )dt

(1.18)
The Doppler spread s f2 is de ned as the second centered moment of the Doppler spectrum: s2 f
fSH ( f )df    S ( f )df SH ( f )df  H  f 2 SH ( f )df
(1.19)
Similarly, the powerdelay pro le is given by SH (t ) = SH ( f , t )df

(1.20)
2 The delay spread s t is de ned as the second centered moment of the powerdelay pro le:
s t2
tSH (t )dt     SH (t )dt  SH (t )dt t 2 SH (t )dt
(1.21)
Since the Doppler spectrum and the time autocorrelation function are Fourier transform pairs, a large Doppler spread s f2 will result in small coherence time Tc and therefore faster temporal fading and vice versa. Similarly, the powerdelay pro le and the frequency autocorrelation function are Fourier transform pairs. Hence, a large delay spread s t2 will result in a small coherence bandwidth Bc and vice versa. In practice, the four parameters are related by Bc and Tc 1 5s f (1.23) 1 5s t (1.22)
1.2.3.3 Frequency Space Transform Mapping. For a static channel, we may extend the time frequency map described in the previous section for the frequency space relationship as illustrated in Figure 1.6. In this diagram, the joint space frequency autocorrelation RH(D , Dr) and the joint delay wavenumber spectrum SH(t, k) are related by Fourier trans
BASIC CONCEPTS IN WIRELESS COMMUNICATIONS
RH (n,r ) Frequency space autocorrelation
Fourier transform pairs
r r=0 RH (n) n=0 RH (r )
Frequency autocorrelation
Spatial autocorrelation
Frequency n
Position r
SH (k)
Doppler spectrum
SH (t)
Wavenumber spectrum
Delay
Wavenumber k
SH (t, k)
Delay wavenumber spectrum
Figure 1.6. Illustration of frequency space autocorrelation and delay wavenumber spectrum.
form pairs. In Section 1.2.2, we have introduced the concepts of coherence distance and angle spread for deterministic channels. We shall try to extend the de nition of these parameters for WSSUS random channels. From the frequency space autocorrelation function, the single dimension spatial autocorrelation of the random channels is given by RH (Dr ) = RH (D , Dr ) D
(1.24)
WIRELESS CHANNEL MODELS
The coherence distance Dc is therefore de ned as the maximum Dr such that RH(Dr) < 0.5. Similarly, we can characterize the statistical behavior of the random channels by the delay wavenumber spectrum SH(t, k). Consider the singledimension wavenumber spectrum SH(k): SH (k) = SH (t , k)dt

(1.25)
2 The angle spread s k is de ned to be the second centered moment of the wavenumber spectrum:
2 sk
kSH (k)dk    S (k)dk (k)dk  H  SH k 2 SH (k)dk
(1.26)
An important indication of the nature of the channel is called the spread factor, given by BcTc. If BcTc < 1, the channel is said to be underspread; otherwise, it is called overspread. In general, if BcTc < 1, the channel impulse < response could be easily measured and the measurement could be utilized at the receiver for demodulation and detection or at the transmitter for adaptation. On the other hand, if BcTc > 1, channel measurement would be extremely > dif cult and unreliable. In this book, we deal mainly with underspread fading channels.
FrequencyFlat Fading Channels
We shall consider the effect of fading channels on a transmitted signal. We rst look at a simple case called a at fading channel. Let x(t) be the lowpass equivalent signal transmitted over the channel and X( ) be the corresponding Fourier transform. The lowpass equivalent received signal y(t, r) is given by y(t , r ) = h(t ; t , r ) x(t  t )dt + z(t ) = H (t ; , r ) X ( ) exp( j 2p t )d + z(t )
 
(1.27) where h(t; t, r) is the timevarying impulse response and H(t; , r) is the timevarying transfer function of the channel. Note that both H(t; , r) and h(t; t, r) are random processes. Suppose that the twosided bandwidth W of x(t) is less than the coherence bandwidth Bc. According to the de nition of the correlation function RH(0; D , 0), the random channel fading H(t, , r) is highly correlated within the range of the transmitted bandwidth [W/2, W/2]. Hence, all the frequency component of X( ) undergoes the same complex fading within the range of frequencies [W/2, W/2]. This means that within the bandwidth W of X( ), we obtain H(t, , r) = h(t; t, r) = h(t, r), where h(t, r) is a complex stationary random process in t and r only. This results in both the