Question

Using FFT to compute linear convolution of real signals. Consider an LTI system with the impulse resonse given below.

h[n] = {sin² (πn/128) cos(πn/5) for 0 ≤ n ≤127
{. 0 otherwise
The system is applied to the input signal given below to produce an output y[n].
x[n] = {mod(n,5) - 5 for 0 ≤ n ≤127
{. 0 otherwise
Using MATLAB or python, compute y[n] for 0≤ n ≤255 using a linear convolution and plot the result. Approximately how many operations are needed to compute the output this way?

Answer 1

To compute the linear convolution of real signals using the FFT, we calculate the Fourier transforms of the input signal and impulse response, multiply them, and take the inverse Fourier transform. Using MATLAB or Python, we can compute the output signal y[n] and plot the result. The approximate number of operations needed for this process is 896.

Step-by-step explanation:

To compute the linear convolution of real signals using the FFT, we first need to calculate the Fourier transform of the input signal and the impulse response. Then, we multiply the Fourier transforms together and take the inverse Fourier transform of the result to obtain the output signal.

In this specific case, we can use MATLAB or Python to perform the calculations. By applying linear convolution and plotting the result, we can get the output signal y[n] for the specified range of n.

To estimate the number of operations needed, we consider the Fast Fourier Transform (FFT) algorithm. The complexity of the FFT is O(N log N), where N is the size of the input signals. In this case, the input signals have a length of 128, so the approximate number of operations needed would be around 128 * log2(128) = 128 * 7 = 896 operations.