State the DFT properties.

1) Periodicity

    If a sequence x(n) is periodic with a periodicity of N samples, then N-point DFT of the sequence is also periodic with a periodicity of N samples.

Hence, if x(n) and x(k) are on N-point DFT pair then

x(n + N) = x(n)      for all n

X(k + N) = X(k)     for all k

2) Linearity

    If x₁(n) ↔ X₁(k) and  x₂(n) ↔ X₂(k) then for any real valued or complex valued constants a₁ and a₂

a₁x₁(n) + a₂x₂(n) ↔ a₁X₂(K) + a₂X₂(K)

3) Parseval's Theorem

    For complex-valued sequences x(n) and y(n)

If x(n) ↔ X(k) and y(n) ↔ Y(k) then ,

∑n=0N−1x(n).y^*(n)=1N∑k=0N−1X(k).Y^*(k)

If x(n) = y(n),

∑n=0N−1|x(n)|2=1N∑k=0N−1|X(k)|2

4) Time Reversal
    If x(n) ↔ X(k) then x((-n)_N = x(N-n) ↔ X((-k))_N = X(N-k)
Hence, reversing the N-point sequence in time is equivalent to reversing the DFT values.

5) Symmetry
    Let assume that N-point sequence x(n) and its DFT are complex valued then,

x(n) = x_R(n) + j x_I(n)    0 ≤  n ≤  N-1
X(k) = x_R(k) + j X_I(k)    0 ≤  k ≤  N-1

where,

xR(n)=1N∑n=0N−1[XR(k)cos(2πknN)−XI(k)sin(2πknN)]

xI(n)=1N∑n=0N−1[XR(k)sin(2πknN)+XI(k)cos(2πknN)]

Similarly,

XR(k)=∑n=0N−1[xR(n)cos(2πknN)+xI(n)sin(2πknN)]

XI(k)=−∑n=0N−1[xR(n)sin(2πknN)−xI(n)cos(2πknN)]

6) Time Shift

    If x(n) ↔ X(k) then

x(n - l)_N ↔ X(k) e^(-j2πkl)/N

7) Frequency Shift
    If x(n) ↔ X(k) then

X(k) e^(j2πkl)/N ↔ X((k-l))_N

8) Convolution (Multiplication of 2 DFT's)

    x(n).y(n) ↔ X(k).Y(k)