4. Lets add an echo to the signal with a delay ofto= 0.12 seconds and an amplitude ofα= 0.8.What is the delayNin samples? Based on the difference equation, create an echo version ofthe signalxusing the commandy = conv(h,x);. What ish?Listen to the output of the difference equation. Do you hear the echo?5.Remove the Echo.The difference equation above for modeling the echo is an LTI system. To remove the echo, weneed to implement the inverse system. What is the impulse response of the inverse system?What is the duration of this impulse response?6. For the echo parameters you used above, plot the impulse response of the inverse system.Make the horizontal axis in your plot in units of SECONDS.7. Implement the inverse of the echo system to performecho cancellation.You can use thecommandg = filter(b,a,y). What are the vectorsbanda?8. Listen to the result. Is the echo removed?9.Estimating Unknown Echo ParametersIn a real problem, you will probably not know what the echo parameterstoandαare. Youwill need to estimate them from the echo signal itself.Suppose you are given the datay(n) (which is corrupted by an echo) but suppose you donot know the value of the delayN.Determine a method of estimatingNbased on theautocorrelation function ofy(n). Letryy(n) be the autocorrelation ofy(n),ryy(n) =y(n)*y(-n),and letrxx(n) be the autocorrelation ofx(n),rxx(n) =x(n)*x(-n).First, find a formula for the signalryy(n) in terms ofrxx(n) (this is not a Matlab question).To do this, use the expression you have for the impulse response together with properties ofthe convolution sum.10. Compute in Matlab and plotrxx(n) vs time (in SECONDS).rxx(n) is the autocorrelation ofthe echo-free signal. To compute in Matlab the autocorrelation of a signal you can use thecommands
I. Selesnick
NYU-Poly
32

11. For the signal and echo parameters above, plotryy(n). Can you determine what the value ofNis from the plot ofryy(n)? (Look at the peaks ofryy(n).) Does it agree with the knownvalue?12. From the course website, there is a data file you can download for this lab.It contains asignal which is corrupted by an echo with unknown echo parameterstoandα.Can you estimateNfor this data from its autocorrelation?13. The amplitudeαis harder to estimate thanN. Once you have estimatedN, try to removethe echo by trying out different values ofαand listening to your result. WhatNandαdoyou find work best? Are you able to successfully remove the echo?14. What if an echo has two components?y(n) =x(n) +α1x(n-N1) +α2x(n-N2)Discuss what system is required for echo cancellation. How would you find the echo parame-ters?
I. Selesnick
NYU-Poly
33

Lab 11: Fourier Series

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