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DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

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DFT (Frequency Domain Sampling)

The Fourier series describes periodic signals by discrete spectra, where as the DTFT describes discrete signals by periodic spectra. These results are a consequence of the fact that sampling on domain induces periodic extension in the other. As a result, signals that are both discrete and periodic in one domain are also periodic and discrete in the other. This is the basis for the formulation of the DFT.

Consider aperiodic discrete time signal x (n) with FT X(w) = DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

Since X (w) is periodic with period 2π , sample X(w) periodically with N equidistance samples with spacing  DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

K = 0, 1, 2…..N-1

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

The summation can be subdivided into an infinite no. of summations, where each sum 
contains 

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

  DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

Put n = n-lN

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

We know that xp(n) =  DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) n= 0 to N-1

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) k=0 to N-1

Therefore   DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) k=0 to N-1

DFT ------------ xp (n) =  DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) n = 0 to N-1

This provides the reconstruction of periodic signal xp(n) from the samples of spectrum 
X(w).

The spectrum of aperiodic discrete time signal with finite duration L<N, can be exactly 

recovered from its samples at frequency Wk= DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

Prove: x(n) = xp (n) 0 ≤ n ≤ N-1

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

Using IDFT

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)  

If we define    p(w) =  DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)  

Therefore: X (w) =  DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)  

At w =2πk/N P (0) =1

And P (w 2πk/N)= 0 for all other values

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

Ex: x(n) = an u(n) 0<a<1

The spectrum of this signal is sampled at frequency Wk= DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) k=0, 1…..N-1, determine reconstructed spectra for a = 0.8 and N = 5 & 50.

X (w) = DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

X (wk) =  DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) k=0, 1, 2… N-1

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) 0≤n≤N-1

Aliasing effects are negligible for N=50

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

If we define aliased finite duration sequence x(n)

xˆ(n)= xp(n)        0≤n≤N-1

= 0                  otherwise

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)   DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

∴Although XÆ (w) ≠ X (w), the samples at Wk= DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) are identical.

Ex: DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) & DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)  

Apply IDFT 

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) using Taylor series expansion

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

= 0  except r = n+mN

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

The result is not equal to x (n), although it approaches x (m) as N becomes ∞ .

Ex: x (n) = {0, 1, 2, 3} find X (k) =?

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) = -2+2j

X (2) = -2
X (3) = -2-2j

DFT as a linear transformation

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) k = 0 to N-1

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) n = 0, 1…N-1

Let xNDFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) XN = DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

The N point DFT may be expressed in matrix form as 

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

Ex: x (n) = {0, 1, 2, 3}

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

IDFT

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

Q.

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

Find y (n) = x (n) h (n) using frequency domain. Since y (n) is periodic with period 2.
Find 2-point DFT of each sequence.

X (0) = 1.5 H (0) = 1.5

X (1) = 0.5    H (1) = -0.5

Y (K) = X (K) H (K)

Y (0) = 2.25 Y (1) = -0.25

Using IDFT  y (0) = 1; y (1) = 1.25

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

= 1 * 0.5 + 0.5 * 1 = 1

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

= 1 * 1 + 0.5 * 0.5 = 1.25

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

 1 * 0.5 + 0.5 * 1 = 1

~y (n) ={1, 1.25, 1, 1.25…..}

Q. Find Linear Convolution of same problem using DFT

Sol. The linear convolution will produce a 3-sample sequence. To avoid time aliasing we convert the 2-sample input sequence into 3 sample sequence by padding with zero.

For 3- point DFT

X (0) = 1.5           H (0) = 1.5

X (1) = 1+0.5  DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) H (1) = 0.5+  DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

X (2) = 1+0.5 e DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE) H (2) = 0.5+ eDFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

Y (K) = H (K) X (K)

Y (0) = 2.25

Y (1) = 0.5 + 1.25 DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)+ 0.5 eDFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

Y (2) = 0.5 + 1.25DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)+ 0.5 e  DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

Compute IDFT

DFT (Frequency Domain Sampling) - Digital Signal Processing, Engineering Notes - Computer Science Engineering (CSE)

y(0) = 0.5

y(1) =1.25

y(2) =0.5

y(n) = { 0.5, 1.25, 0.5} Ans 

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