Question

We look at echo cancellation, which is a problem that occurs quite frequently. The signal that we observe is corrupted by one or more echoes, which are just delayed and scaled copies of the original signal. The goal of echo cancellation is to process the corrupted signal, and recover the original. There are two types of systems where echo cancellation is applied. In the first case, there are only a few echoes. This occurs in broadcasting when there are several different transmission paths between the broadcast antenna and your receiving antenna, each with a different path length, path gain, and delay. This is the multipath problem. You have seen this if you watched broadcast TV in a low-signal area. Each unique transmission path contributes a ghost image, shifted in the scan direction. Here is the case of a primary path and one additional transmission path (echo), with the impulse response

h[n]=δ[n]+αδ[n−Nd]
where Nd is the delay between the primary path and the echo, and α is the size of the echo. This system is described by the difference equation
y[n]=x[n]+αx[n−Nd]
The other type of echo cancellation system deals with the case where there is an infinite sequence of decaying echoes in response to an input. This occurs in the acoustics of a concert hall where the sound is reflected off a series of surfaces. The impulse response for this system is
h[n]=δ[n]+αδ[n−Nd]+α2δ[n−2Nd]+α3δ[n−3Nd]+⋯=∑m=0+[infinity]αmδ[n−mNd]. The difference equation that corresponds to this impulse response is
y[n]=αy[n−Nd]+x[n] This is the echo-generation system that we will be concerned with in this lab. The echo-cancellation system for this case is z[n]=y[n]−αy[n−Nd] where z[n] is the signal with the echoes removed. All we need to do to suppress the infinite train of echoes is to subtract a scaled and delayed copy of y[n].

Task 1.
System description We first look at the overall block diagram of the system, and show that the overall difference equation suppresses the echoes.
Draw a block diagram of the echo-generation system followed by the echo-cancellation system.
Derive the overall difference equation for the cascade of the echo-generation system followed by the echo-cancellation system. Does this actually cancel the echoes, i.e. is z[n]=x[n] ?

Answer 1

The question involves the engineering concept of echo cancellation using an echo-generation system followed by an echo-cancellation system, aiming to restore the original signal by subtracting scaled, delayed copies of it. It is analogous to destructive interference, which is used in noise reduction applications.

Step-by-step explanation:

The question pertains to echo cancellation in signal processing, a concept in engineering where the goal is to remove delayed and scaled copies, or echoes, from an observed signal. The signal is expressed by the difference equation y[n]=\alpha y[n-Nd]+x[n], which stands for the echo-generation system. The echo-cancellation system is subsequently defined by z[n]=y[n]-\alpha y[n-Nd]. To demonstrate echo cancellation, one would combine both systems in a cascade and prove that z[n] is indeed the original signal x[n].

When analyzing this cascade, if we substitute the equation for y[n] into the equation for z[n], we would observe that the echoes generated by the first system are being canceled by the second system. The result of the cancellation would ideally lead to z[n]=x[n], effectively restoring the original signal.

This concept can be compared to noise reduction through destructive interference, where sound interference is used to cancel noise, such as in soundproofing applications within commercial aircraft. The principle of destructive interference uses the principle of superposition from physics to combine two waves in such a way as to cancel each other out when they are out of phase.