## 5.3. Exercises

EXERCISE 5.14.

Consider the following digital signals:

Demonstrate analytically that the real parts of the DFT_{1D} of the signals *x[n*] and *y*[*n*] are equal. Verify this property using a MATLAB code.

EXERCISE 5.15.

Using a MATLAB code, verify the Parseval theorem:

for the following digital signal:

EXERCISE 5.16.

The function ρ^{i} mod *N*, *i* [1, *N* −1] performs the permutation of the sequence {1,2,3,..., *N* −1} into the sequence {ρ^{1}, ρ^{2}, ρ^{3},..., ρ^{N−1}} if *N* is a prime number and ρ is a primitive root of *N*. For example, the permutation of the sequence {1, 2, 3, 4} into the sequence {2, 4, 3, 1} is obtained for *N* = 5, as it is illustrated in the table below.

The sequence {1,2,3,. *N* −1} can be also permutated into the sequence {ρ^{i} mod *N* with *i* = 0..*N* −2}. Thus, in the case *N* = 5 and ρ = 2, the following permutation is obtained:

In the Galois field GF(5), the element ρ = 2 is a primitive ...

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