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State any five DFT properties.

Mumbai University > Computer Engineering > Sem 7 > Digital Signal Processing

Marks: 10 Marks

Year: May 2016

1 Answer
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1. Shifting Property

If $x(n) \leftarrow FT\rightarrow x(k) OR x(n) \leftarrow FT\rightarrow X(\omega)$

Then, $x(n-m)\leftarrow FT\rightarrow W_N^{mk}.x(k)$

i.e, $x(n-m)\leftarrow FT\rightarrow e^{-j\omega k } X(\omega)$

& $x(n+m) \leftarrow FT\rightarrow W_N^{-mk}.x(k)$

Shifting property states that when a signal is shifted by m samples then the magnitude spectrum is unchanged but the phase spectrum is changed by amount $(-\omega k)$.

2. Frequency Shifting

$W_N^{mn}.x(k) \leftarrow FT\rightarrow x(k+m) \\ W_N^{-mn}.x(k) \leftarrow FT\rightarrow x(k-m)$

3. Conjugate Property

$x(n) \leftarrow FT\rightarrow x(k) \\ x*(n) \leftarrow FT\rightarrow x*(-k)$

4. Symmetric Property

$x(n) \leftarrow FT\rightarrow x(k)$

If x(n) = x(-n)

Then x(k) = x*(N-k)

5. Convolution Property

If, $x_1(n) \leftarrow FT\rightarrow x_1(k) \& x_2(n) \leftarrow FT\rightarrow x_2(k)$

Then, $x_1(n) * x_2(n) \leftarrow FT\rightarrow x_1(k).x_2(k)$

Convolution of two signals in time domain is equivalent to multiplication in frequency domain.

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