Test: Frequency Domain Sampling

Explanation: If x(n) is a finite duration sequence of length L, then the Fourier transform of x(n) is given as

If we sample X(ω) at equally spaced frequencies ω=2πk/N, k=0,1,2…N-1 where N>L, the resultant samples are

Explanation: If X(k) discrete Fourier transform of x(n), then the inverse discrete Fourier transform of X(k) is given as

Explanation: The Fourier transform of this sequence is

If N=L, then X(k)= L for k=0
=0 for k=1,2….L-1

Explanation: We know that the Discrete Fourier transform of a signal x(n) is given as

Thus we get Nth rot of unity WN= e^-j2π/N

Explanation: The formula for calculating N point DFT is given as

From the formula given at every step of computing we are performing N complex multiplications and N-1 complex additions. So, in a total to perform N-point DFT we perform N² complex multiplications and N(N-1) complex additions.

Explanation: If XN represents the N point DFT of the sequence xN in the matrix form, then we know that

Explanation: The first step is to determine the matrix W4. By exploiting the periodicity property of W4 and the symmetry property
W_N^k+N/2= -W_N^k
The matrix W₄ may be expressed as

Explanation: The Fourier series coefficients are given by the expression

  Given x(n)={0,1,2,3}
We know that the 4-point DFT of the above given sequence is given by the expression

In this case N=4
=>X(0)=6,X(1)=-2+2j,X(2)=-2,X(3)=-2-2j.

Explanation: We know that according to the periodicity and symmetry property,
100/4=200/x=>x=8.