Question

Consider a real LTI system with the following properties. 1 - Its frequency response H(w) is real and linear on [0, 1] (and therefore on ( - 11,0] by symmetry) as illustrated below (warning: the true values and signs of Q and b may not be those shown in the graph which is only for illustration). 2. It preserves the input ( - 1)". 3- It transforms the periodic sequence X[n]shown below into a constant sequence. What must be the value of this constant ? 3 3 H(w) 3 ac[n] 2 2 2 2 2 2 9 a O o -5-4-3-2-1 0 1 2 3 4 5 n w O-2 2 There is not enough information to answer this question. None of the above

Answer 1

Through inspection notice, the filter lets lower frequencies to go through, a low pass filter.

The offset of x[n] is 2 that is the 0 frequency value, after filtering the function its expected to obtain the lower frequency component, in this case 2.

And the train of pulses x[n] can be defined through a sum of functions in the form of (-1)^n.

The above train of pulses can be defined by the following two functions:

And the function

So the function x[n] is:

Lets transform (-1)^an to an exponential equivalent:

Notice the speed at which the exponential covers the complex plane is:

Thus, the frequency of each component of the function x[n] is .

- For x1[n].

The frequency of these terms is:

For x2[n].

Finally, Looking at , notice how frequencies above are neglected, and has the greatest amplitude.

Since will be a function of the frequency components of X:

We expect the component at to be almost 0, due to the attenuation of the filter at this frequency:

And the component at to be preserved:

A constant sequence of 2