Question: In many practical situations a signal is recorded in the presence of an echo, which we would like to remove by appropriate processing. For example, in Figure P7.41 (a), we illustrate a system in which a receiver simultaneously receives a signal x(t) and an echo represented by an attenuated delayed replication of x(t). Thus, the receiver output is
Assume that x(t) is band limited [i.e., X(jω) = 0 for
(b) With the assumptions of part (a), specify the gain A of the ideal lowpass filter such that
(c) Now suppose that
(a) The received signal is,
It is ideally sampled with an impulse train of period,
Sampling with impulse functions is ideal sampling.
Translate the received signal to frequency domain.
The sampled signal in time domain is (or) the impulse train is,
It is a sequence of impulses with amplitudes equal to the sample value.
Remove the sampling period, as they simply correspond to a sample.
Convert the impulse train to a sequence.
Therefore, the sequence is,
The final output must be proportional only to
That implies the signal
For the final output to be proportional to be only
That means in the delayed version of
In frequency domain, this translates to elimination of the
So the digital filter
The expression for output
Convolution in time domain translates to multiplication in frequency domain.
Thus, the frequency domain description of the digital filter is,
Determine the difference equation.
Apply inverse Fourier transform in equation (1).
Thus, the difference equation of the filter is
The output sequence
That is, the sequence itself is an impulse train.
To recover the signal form this impulse train, the signal is passed through a low pass filter.
The output of the digital filter is taken to be exactly equal to .
Thus, the input to the low pass filter is
The low pass filter simply acts as a reconstruction filter.
The input is
Thus, the pass band gain of the ideal low pass filter is
Note that in this case,
The minimum sampling frequency, to avoid aliasing is,
The delay of the echo has a range of,
The sampling frequency of the sum of two signals is twice the bandwidth of the signal with the highest bandwidth.
The discrete sequence is,
The sampling frequency does not change if the signal is shifted.
The received signal is the sum of the signal
The sampling frequency of sum of two signals is the twice the bandwidth of the signal with the highest bandwidth.
Thus, the sampling frequency is,
Thus, the sampling period must be ,
Apply Fourier transform to equation (1).
Determine the frequency response of the digital filter,
For the final output to be proportional to,
The only possibility is altering the function using
The input signal is,
Since the input is directly applied to the digital filter,
The frequency response of the output is multiplication of
Thus, the gain of frequency response of the digital filter is,
The impulse train
For the final output to be exactly
The input to the ideal low pass filter is a sampled version of
So get the exact replica of the signal
Thus, the gain of the ideal low pass filter is