Time Varying dimension: To characterize the time variation dimension, we can
have either the Doppler spread or coherence time. Doppler spread [ 1671 refers
to the spread of frequency introduced by the fading channel when we transmit
a single tone signal (narrow pulse in frequency domain) as illustrated in
Figure 1.7(a). The spread in frequency is due to the mobility between the
transmitter and the receiver or the mobility of the surrounding obstacles. The
maximum Doppler spread is given by:
where ƛ is the wavelength of the signal and ti is the maximum speed between
the mobile and the base station. Typical values of Doppler spread for pedestrian
users (5km/hr) and vehicular users (100km/hr) at carrier frequency of 2.4 GHz
are given by 14 Hz and 300 Hz respectively.
Figure 1.6 Illustration of the delay spread and coherence bandwidth.
Figure 1.7 Illustration of the doppler spread and coherence time.
Similarly, an equivalent parameter to quantify the time variation dimension of
microscopic fading is given by the coherence time. Figure 1.7(b) illustrates the
concept of coherence time. Suppose we transmit two narrow pulses in time
domain at times t = tl and t = t 2 with unit energy. After channel fading, the
corresponding signals will be received at t = tl + 71 and t = t 2 + 7 2 with
energies a1 and a2 respectively. Similar to coherence bandwidth, if we separate
the two transmissions sufficiently far in time, (│ ț1 +ț2 │z > Tc), the channel
fading a1 and a2 are uncorrelated. Hence, the coherence time is defined as
the minimum time separation at the transmitter in order to have uncorrelated
fading at the receiver. It is found that for 0.5 correlation coefficient between
a1 and a2, the coherence time T, is related to the Doppler spread f d by:
Note that Doppler spread and coherence time are two sides of the same coin to
characterize the time variation dimension of microscopic fading.
1.4.1 Fiat Fading vs Frequency Selective Fading
Based on the delay spread and the transmitted symbol duration, (or equivalently,
the coherence bandwidth and the transmitted bandwidth), the multipath fading can
be classified as frequencyjat fading or frequency selective fading [287]. A fading
channel is said to be frequencyjat if ~7~< T, or equivalently, B, > Wt, where T,
and W,, are the symbol duration and the transmitted signal bandwidth, oT and B, are
the delay spread and the coherence bandwidth, respectively. Otherwise, the fading
channel is said to be frequency selective. Note that whether the transmitted signal
will experience frequency flat fading or frequency selective fading depends on both
the channel parameters (such as oT,B ,) as well as the transmitted signal parameters
(such as T, and Wt,).
When a transmitted signal experience frequency flat fading, the received signal
consists of superposition of multipath signals but the multipaths have delays much
smaller than the symbol duration T, and hence, the multipaths are unresolvable. As
a result of the unresolvable multipath, the received pulse shape will be different from
the transmitted pulse shape but the received pulse appears as one single pulse at the
receiver. The received signal is modeled as:
where h(t) is the time-varying channel fading which is modeled as complex Gaussian
random variable with zero mean and unit variance, z ( t ) is the complex AWGN noise
and z ( t ) is the low-pass equivalent complex transmitted signal.
On the other hand, when a transmitted signal experience frequency selective fading,
the received signal consists of multipaths with delay >> symbol duration (T,). Hence,
the received signal consists of pulses at resolvable deZays (integral multiples of T,),
In general, the number of resolvable multipaths is given by:
The received signal for frequency selective fading channel is modeled as:
where hi ( t )is the time varying channel fading for the i-th resolvable multipaths which
is modeled as complex Gaussian random variable with zero mean and unit variance.
to be continue....