820 Consider a baseband M-ary system using M discrete am-plitude levels. The receiver model is as shown in Figure P8.20; the operation of which is governed by the following assumptions: (a) The signal...

Solution:
a) Given signal is shown below.
The impulse response of the matched filter is ( ) ( ) and it is shown below.
) The output of the matched filter is the convolution of h (t) with s (t). It is shown
elow.
c) The peak value of the filter output is equal to


, which occurs at .
Solution: Consider a binary PCM system in the presence of channel noise, the receiver is
shown in figure below.
The average probability of e
or for a system is considered under the following assumptions.
i. The PCM system uses an on-off format, in which symbol 1 is represented by A
volts and symbol 0 by zero volt.
ii. The symbols 1 and 0 occur with equal probability.
iii. The channel noise w (t) is white and Gaussian with zero mean and power spectral
density No/2.
The average probability of e
or is determined considering two possible kinds of e
ors.
The first kind of e
or occurs when symbol 0 is sent and the receiver chooses symbol 1. In
this case, the probability of e
or is just the probability that the co
elator output will exceed
the threshold λ owing to the presence of noise, so the transmitted symbol 0 is mistaken for
symbol 1. Since the apriori probabilities of symbols 1 and 0 are equal, we have P0 = P1.
The expression for the threshold λ will be




where is the bit duration, and
is the signal energy consumed in the transmission of
symbol 1.
The co
elator output denoted by ‘y’ is given by
∫ ( ) ( )
Under hypothesis H0, co
esponding to the transmission of symbol 0, the received signal x(t)
equals the channel noise w (t). We may therefore describe the co
elator output as
∫ ( )
Since the white noise w (t) has zero mean, the co
elator output under hypothesis H0 also
has zero mean. In such a situation, a conditional mean is considered which is written as
( | ) [∫ ( )
]
Where the random variable Y represents the co
elator output with y as its sample value
and W(t) is a white noise process with w(t) as its sample function. The subscript 0 in the
conditional mean refers to the condition that hypothesis is true. Co
espondingly, let

denote the conditional variance of the co
elator output, given that hypothesis is
true. Therefore

( | ) ∫ ∫ ( )
( )
The double integration accounts for the squaring of the co
elator output. Interchanging the
order of integration and expectation, we get

∫ ∫ ( )
( )
∫ ∫
( )
The parameter ( ) is the ensemble averaged autoco
elation function of the white
noise process W (t). From random process theory, it is recognized that the autoco
elation
function and power spectral density of a random process form a Fourier Transform pair.
Since the white noise W (t) is assumed to have a constant power spectral density of N0/2, it
follows that the auto co
elation function of such a process consists of a delta function
weighted by N0/2.
This can be written as
( )


( )
Substituting ( ) in
and considering the property that the total area under the
Dirac delta function ( ) is unity, we get






The statistical characterization of the co
elator output is computed by noting that it is
Gaussian distributed, since the white noise at the co
elator input is itself Gaussian.
Therefore under hypothesis H0 the co
elator output is a Gaussian random variable with
zero mean and variance

and is given by
( )

√
(


)
Where the subscript in ( ) signifies the condition that symbol 0 was sent.
The above figure shows the bell shaped curve for the probability density function of the
co
elator output, given that symbol 0 was transmitted. The probability of the receiver
deciding in favour of symbol 1 is given by the shaded area in the figure. The part of the y-
axis covered by this area co
esponds to the condition that the co
elator output y is in
excess of the threshold λ above.
Let P10 denote the conditional probability of e
or, given that symbol 0 was sent. Therefore
∫ ( )


√
∫ (


)
Considering

√



√
∫ ( )
√
Complementary e
or...