An LTI system is described by the difference equation
y[n]=x[n]-2x[n-1] x[n-2]
. Its output signal
y[n]
approximates computation of the second derivative of the input signal
x[n]
.
i) Calculate and sketch the system impulse response
h[n]
(recall that
h[n]=y[n]
when
x[n]=\delta [n]
). Is this a causal system? Is it (BIBO) stable? State reasons for your answers.
ii) Consider the input signal
x[n]=[1,4,9]
(in this vector representation of
x[n]
, the underlined sample is the sample at time index
n=0
; sample values at other time indices not listed have zero values). Calculate the output
y[n]=x[n]^(**)h[n]
over
-1<=n<=6
, where ***denotes linear convolution. Use a tabular approach to evaluate the convolution. Over what time indices
n
is the output
y[n]
neither transient nor end effect? Briefly explain,
iii) Compute
y[n]=h[n]^(**)x[n]
numerically using the Matlab function 'conv' and check your answer to part (ii) above. Is the length of
y[n]
as expected? Explain. [Hint: type "help conv" in the Matlab command window to learn more about the input and output arguments of the "conv" function].