ECE 361: Fundamentals of Digital Communications
Lecture 23: Frequency Diversity
Introduction
In many instances, temporal diversity is either not available (in stationary scenarios) or
cannot be efficiently harnessed due to the strict delay constraints of the data being commu-
nicated. In these instances, and as an added source of diversity, looking to the frequency
domain is a natural option.
Frequency Diversity Channel
Frequency diversity occurs in channels where the multipaths are spread out far enough,
relative to the sampling period, so that multiple copies of the same transmit symbol are
received over different received samples. Basically, we want a multitap ISI channel response:
the L-tap wireless channel
y[m] =
L−1
X
`=0
h`x[m−`] + w[m], m ≥1 (1)
is said to have L-fold frequency diversity. The diversity option comes about because the
different channel taps h0, . . . , hL−1are the result of different multipath combinations and
are appropriately modeled as statistically independent. Thus the same transmit symbol (say
x[m]) gets received multiple times, each over a statistically independent channel, (in this
case, at times m, m + 1, . . . , m +L−1).
A Single Bit Over a Frequency Diversity Channel
Suppose we have just a single bit to transmit. The simplest way to do this is to transmit
the appropriate symbol x=±√Eand stay silent. : i.e., we set
x[1] = ±√E, (2)
x[m]=0 m > 1.(3)
At the receiver, we have L(complex) voltages that all contain the same transmit symbol
immersed in multiplicative and additive noises:
y[`+ 1] = h`x+w[`+ 1], ` = 0, . . . , L −1.(4)
As before, we suppose coherent reception, i.e., the receiver has full knowledge (due to accurate
channel tracking) of the channel coefficients h0, . . . , hL−1. We see that the situation is entirely
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