Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE) PDF Download

We know that when ω = 2πK/N and N→∞,ω becomes a continuous variable and limits summation become −∞ to+∞.

Therefore,

Discrete Time Fourier Transform (DTFT)

We know that,

Where,   is continuous and periodic in ω and with period 2π     .…eq(1)

Now,

        … From Fourier series

ω becomes continuous and   because of the reasons cited above.

    …eq(2)

Inverse Discrete Time Fourier Transform

Symbolically,

   (The Fourier Transform pair)

Necessary and sufficient condition for existence of Discrete Time Fourier Transform for a non-periodic sequence x(n) is absolute summable.

Properties of DTFT

• Linearity :  
• Time shifting − 
• Time Reversal −  
• Frequency shifting − 
• Differentiation frequency domain − 
• Convolution − 
• Multiplication −  
• Co-relation − 
• Modulation theorem −
• Symmetry −
  
• Parseval’s theorem − 

Earlier, we studied sampling in frequency domain. With that basic knowledge, we sample X(e^jω) in frequency domain, so that a convenient digital analysis can be done from that sampled data. Hence, DFT is sampled in both time and frequency domain. With the assumption x(n)=x_p(n)

Hence, DFT is given by −

    …eq(3)

And IDFT is given by − 

    n = 0,1,….,N−1  …eq(4)

Twiddle Factor

It is denoted as W_N and defined as    Its magnitude is always maintained at unity. Phase of  . It is a vector on unit circle and is used for computational convenience. Mathematically, it can be shown as −

• It is function of r and period N.

  Consider N = 8, r = 0,1,2,3,….14,15,16,….

  
• 

Linear Transformation

Let us understand Linear Transformation −

We know that,

Note − Computation of DFT can be performed with N² complex multiplication and N(N-1) complex addition.

•  N point vector of signal x_N
•  N point vector of signal X_N
  
  
  N - point DFT in matrix term is given by - XN = W_Nx_N

  Matrix of linear transformation

IDFT in Matrix form is given by

Comparing both the expressions of  and  

Therefore, W_N is a linear transformation matrix, an orthogonal (unitary) matrix.

From periodic property of W_N and from its symmetric property, it can be concluded that, 

Circular Symmetry

N-point DFT of a finite duration x(n) of length N≤L, is equivalent to the N-point DFT of periodic extension of x(n), i.e. x_p(n) of period N. and   . . Now, if we shift the sequence, which is a periodic sequence by k units to the right, another periodic sequence is obtained. This is known as Circular shift and this is given by,

The new finite sequence can be represented as

Example − Let x(n)= {1,2,4,3}, N = 4,

x′_p(n) = x(n−k, modulo N) ≡ x((n−k)) _N; ex − if k = 2i.e 2 unit right shift and N = 4,

Assumed clockwise direction as positive direction.

Conclusion − Circular shift of N-point sequence is equivalent to a linear shift of its periodic extension and vice versa.

Circularly even sequence −

Conjugate even − 

Circularly odd sequence − 

Conjugate odd −  

Now, 

For any real signal x(n),

Time reversal − reversing sample about the 0^th sample. This is given as;

Time reversal is plotting samples of sequence, in clockwise direction i.e. assumed negative direction.

Some Other Important Properties

Other important IDFT properties x(n)⟷X(k)

• Time reversal 
• Circular time shift  
• Circular frequency shift 
• Complex conjugate properties −
  
• Multiplication of two sequence −
  
• Circular convolution − and multiplication of two DFT
      m = 0,1,2,....,N - 1
  
• Circular correlation −   , then there exists a cross correlation sequence denoted as   such that 
• Parseval’s Theorem −