Output y[n] from G[k]=X[k]H[k]

Here G[k] = X[k]H[k] with 4-point DFTs, so the output y[n] = IDFT{G[k]} is the 4-point circular convolution of x[n] and h[n] (multiplication of DFTs ⇔ circular convolution). Both sequences already have N = 4 points, so no zero padding is needed:

• x[n] = {1, 2, 3, 4}
• h[n] = {1, 3, 5, 7}

y[n] = Σ (m = 0 to 3) x[m] · h[(n − m) mod 4]

Written as a matrix product (the first column of the circulant matrix is x, and each later column is the previous one shifted down circularly):

|    1    4    3    2 |   | 1 |
|    2    1    4    3 |   | 3 |
|    3    2    1    4 |   | 5 |
|    4    3    2    1 |   | 7 |

Row by row:

• y[0] = (1)(1) + (4)(3) + (3)(5) + (2)(7) = 42
• y[1] = (2)(1) + (1)(3) + (4)(5) + (3)(7) = 46
• y[2] = (3)(1) + (2)(3) + (1)(5) + (4)(7) = 42
• y[3] = (4)(1) + (3)(3) + (2)(5) + (1)(7) = 30

Check using DFTs: the inverse DFT of X(k)·H(k) for k = 0 … 3 is {42, 46, 42, 30}. Also, Σy[n] = 160 = X(0)·H(0) = 10 × 16 = 160.

Answer: y[n] = {42, 46, 42, 30}, n = 0 … 3.