Question

1. A length-10 discrete-time signal x[n] is defined by the values { 6, 1, 1, 6, 1, 1, 6, 1, 1, 0 }.
After taking the 10-pt DFT, X[k], determine the DC value, i.e., X[0].

2. Suppose that the N-point DFT of the signal x[n]x[n] is X[k]. A new signal y[n] is formed by adding a constant:

y[n]=x[n]+1 for n=0,1,2,…,N−1n=0,1,2,…,N−1

The N-point DFT of Y[k] is:

Answer 1

Part 1.

To calculate DFT of N time, the formula is

where N = 10, and 0<=k<=10

for DC value of X(k) can be calculated by putting k = 0 in the equation

  

  

  

  

Part 2

given us

  

which means at every time instant we are adding a constant 1

therefor y(n) become y(n) = {7,2,2,7,2,2,7,2,2,1}

now form the dft eqution

  

for k = 0;

  

  

  

for k = 1;

  

  

for k = 2;

  

  

similarly

  

  

  

  

  

  

  

we also can use the 10*10 square DFT matrix to calculate 10 points DFT, but it will be quite complex.