Question

A real-valued discrete-time signal x[n] has a Fourier transform X(eʲω) that is zero for 3π/14≤∣ω∣≤π. The nonzero portion of the Fourier transform of one period of X(eʲω) can be made to occupy the region ∣ω∣<π by first performing upsampling by a factor of L and then performing downsampling by a factor of M. Specify the values of L and M.

Answer 1

To make the nonzero portion of the Fourier transform occupy the region ∣ω∣<π, we need to perform upsampling by a factor of L and then downsampling by a factor of M. The values of L and M that satisfy the equation are L = π - 3π/14 and M = π.

Step-by-step explanation:

To make the nonzero portion of the Fourier transform occupy the region ∣ω∣<π, we need to perform upsampling by a factor of L and then downsampling by a factor of M. Upsampling increases the number of samples, and downsampling decreases the number of samples.

Let's assume the original signal x[n] has N points. After upsampling by a factor of L, the signal will have L*N points. Then, after downsampling by a factor of M, the signal will have L*N/M points.

To make the nonzero portion occupy ∣ω∣<π, the number of points L*N/M should be equal to the number of points in the range 3π/14≤∣ω∣≤π. Therefore, we can set up the equation:

L*N/M = (π - 3π/14)*N/π

Simplifying the equation, we get:

L/M = (π - 3π/14)/π

So, the values of L and M that satisfy the equation are L = π - 3π/14 and M = π.