Channel Autocorrelation Functions and Power Spec-

Autocorrelation Functions

We assume c(t; τ ) is wide-sense-stationary (WSS) and define the autocorrelation func-tion of c(t; τ ) as

R_c(∆t; τ₁, τ₂) = E[c(t; τ₁)c^∗(t + ∆t; τ₂)]. (2.9) In most cases, the attenuation and phase shift at the delay τ₁ path is uncorrelated with that at τ2, which is known as uncorrelated scattering (US). With the WSSUS condition,

(2.9) can be decoupled into

R_c(∆t; τ₁, τ₂) = R_c(∆t; τ₁)δ(τ₂− τ₁). (2.10)

Notice that R_c(0; τ ) , Rc(τ ) is called the power delay profile (PDP); the range of τ within which Rc(τ ) is non-zero is termed the maximum delay spread of the channel and is denoted as T_m.

PDP describes the average received power as a function of delay and is one of the most important parameters for channel modeling. We will see later that many indus-trial standards specify PDPs in their testing environments. PDP can be measured by probing the channel with a wideband radio-frequency (RF) waveform that is generated by modulating a high-rate pseudo-noise (PN) sequence. By cross correlating the re-ceiver output against delayed versions of the PN sequence and measuring the average value of the correlator output, one can obtain the power versus delay profile. Just like there are may equally valid definitions of bandwidth, other useful measurements of the delay spread are possible. One of them is the root-mean-square (RMS) delay spread, which is defined by

T_RMS =

sR τ²Rc(τ )dτ

R Rc(τ )dτ − (R τ Rc(τ )dτ

R Rc(τ )dτ )². (2.11)

Now consider channel characterizations in the frequency domain. By taking the Fourier transform of c(t; τ ) w.r.t. the variable τ , the time-variant channel frequency response is

C(t; f ) = Z ∞

−∞

c(t; τ )e^{−j2πf τ}dτ. (2.12) Similarly, we can define the autocorrelation function of C(t; f ) as

R_C(∆t; f₁, f₂) = E[C(t; f₁)C^∗(t + ∆t; f₂)]. (2.13)

Relating (2.13) to (2.10), it can be shown that

R_C(∆t; f₁, f₂) = Z ∞

−∞

R_c(∆t; τ₁)e^{−j2π∆f τ1}dτ₁ , RC(∆t; ∆f )

(2.14)

where ∆f = f₂ − f₁. Equation (2.14) describes the autocorrelation function in the frequency variable. Moreover, the range of ∆f within which the components of RC(∆f ) are highly correlated is defined as the coherence bandwidth of the channel and denoted as (∆f )_c. As R_c(∆t; τ₁) and R_C(∆t; ∆f ) form a Fourier transform pair, a very rough relation is that the coherence bandwidth is reciprocally proportional to the maximum delay spread [33]

(∆f )_c≈ 1

T_m. (2.15)

If the signal bandwidth is large compared to the channel’s coherence bandwidth, the signal will be distorted and the channel is called frequency-selective. This is equivalent to the case where the delay spread is larger than the symbol time, which is also termed time-dispersion because transmitting an ideal impulse through the channel will yield a receive signal with several delayed pulses. In this case, the interference among different symbols occur and called inter-symbol interference (ISI).

Power Spectral Density

Now, we consider the time variations of the channel and investigate the Fourier trans-form pair

S_C(λ; ∆f ) = Z ∞

−∞

R_C(∆t; ∆f )e^{−j2πλ∆t}d∆t. (2.16)

If ∆f is set to 0, S_C(λ) is called the channel’s Doppler PSD and λ represents the Doppler frequency. The range of λ within which SC(λ) is non-zero is termed the maximum Doppler spread of the channel and is denoted as f_D. The maximum Doppler

Table 2.1: Time and Frequency Dispersion

Time Dispersion Frequency Dispersion Time Domain Delay spread Time selective fading (fast fading)

Interpretation ISI Coherence time

Frequency Domain Frequency selective fading Doppler spread

Interpretation Coherence bandwith ICI

frequency can be roughly calculated by

f_D = vf_c

c (2.17)

with v being the mobile speed and c the speed of light. Similarly, the range of ∆t within which the components of RC(∆t) are highly correlated is defined as the channel’s coherence time and is denoted as (∆t)_c. Again, because they form a Fourier transform pair, the maximum Doppler spread and the coherence time are reciprocally related via

(∆t)_c≈ 1

f_D. (2.18)

Similarly, if the signal duration is large compared to the channel’s coherence time, the channel is called time-selective. This is equivalent to the case where the Doppler spread is large enough, and a pure-tone transmit signal passing through the channel will yield a receive signal with several frequency components; we call this phenomenon frequency-dispersion. The terminologies and their relationships are summarized in the Table 2.1.

To relate the parameters τ , λ, ∆f , and ∆t, we define the scattering function of the channel

S_sc(λ; τ ) = Z ∞

−∞

R_C(∆t; ∆f )e^{−j2πλ∆t}e^{j2πτ ∆f}d∆t d∆f , (2.19)

which is the double Fourier transform of R_C(∆t; ∆f ).

Jakes’ model is widely adopted to describe the time variation of the mobile radio channels with the the corresponding autocorrelation function

R_C(∆t) = J₀(2πf_D∆t) (2.20)

where J0(·) is the zero-th order Bessel function of the first kind. The Dopper PSD is obtained by Fourier transform, that is

SC(f ) =

Maxmium Delay Spread = 80 ns

RMS Delay Spread = 15.6 ns

Figure 2.1: PDP of IEEE 802.11n channel model B.

Here we give some examples of the statistical channel parameters in practical systems. For the Wireless Local Area Network (WLAN), which is usually used in the

indoor, there are five channel models proposed by the IEEE 802.11 a/n standard [34]

where the RMS delay spreads are about 0 ns to 150 ns depending on the scenarios, and the Doppler power spectrum is Bell-shaped, specified by

S_C(f ) = 1 1 + A(_f^f

D)². (2.22)

Fig. 2.1 shows the PDP of the IEEE 802.11n channel B whose the maximum delay spread is 80 ns and the RMS delay spread is 15.6 ns. According to (2.15), the co-herence bandwidth is around 12.5 MHz. The channel frequency response is shown in Fig. 2.2 and it can be seen that the signal with 20 MHz bandwidth experience frequency-selectivity while the CFR varies slow within 5 MHz and thus the channel of the signal with 5 MHz bandwidth can be considered as flat-fading. However, since the application is in indoor, the mobility is very small and the channel varies very slowly so we demonstrate the Doppler effect in the following example.

For the wireless Metropolitan Area Network (MAN), e.g., IEEE 802.16e, the Inter-national Telecommunication Union (ITU) channel models are adopted where the RMS delay spreads for outdoor scenarios are around 2 us to 20 us, much longer than that of WLAN scenarios. The Jakes’ U-shape Doppler spectrum is assumed and the typical maximum Doppler spreads at 2.5 GHz carrier frequency are 4.6, 104.2, and 231.5 Hz corresponding to the speed of 2, 45, and 100 km/h, respectively. The corresponding coherence time is 200, 10, and 4 ms, respectively. Fig. 2.3 shows the path gain varia-tions of ITU Vehicular-A channel and it can be seen that higher speeds result in faster channel variations. In IEEE 802.16e, the symbol duration is around 100 µs so that even at 100 km/h the channel still can be considered as linearly time-varying within one symbol.

0 1 2 3 4 5 6 7 8 9 10 x 10⁷ 0

1 2 3 4 5 6

frequency (Hz)

Amplitude

5 MHz BW

20 MHz BW

Figure 2.2: Channel frequency response of the PDP described in Fig. 2.1.

Large-scale versus small-scale fading

It is worthwhile to mention that channel fading can also be categorized into large-scale and scale fading [32]. While we have discussed the causes and effects of small-scale fading in the above, the large-small-scale fading is caused by propagation loss over long distance and by shadowing due to obscuring objects that attenuate the received signal strength. Since large-scale fading varies much slower compared to small-scale fading and the induced issues are more related to cell planning and receivers’ sensitivity, in this dissertation we only focus on small-scale fading caused by multipath propagation.

0 1 2 3 4 5 6 7 8 9 10 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

ms

Magnitude

fD = 4.6 Hz fD = 104.2 Hz fD = 231.5 Hz

Figure 2.3: Path gain variations of ITU Vehicular-A channel with speeds at 2, 45, and 100 km/h.