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,

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

Discrete Time Fourier Transform (DTFT)

We know that, Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

Where,  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE) is continuous and periodic in ω and with period 2π     .…eq(1)

Now,

Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)        … From Fourier series

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

ω becomes continuous and  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE) because of the reasons cited above.

Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)    …eq(2)

Inverse Discrete Time Fourier Transform

Symbolically,

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

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

Properties of DTFT

  • Linearity :  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Time shifting −  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Time Reversal −   Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Frequency shifting −  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Differentiation frequency domain −  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Convolution −  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Multiplication −   Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Co-relation −  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Modulation theoremTime Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Symmetry
    Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Parseval’s theorem −  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

Earlier, we studied sampling in frequency domain. With that basic knowledge, we sample X(e) 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)=xp(n)

Hence, DFT is given by −

Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)    …eq(3)

And IDFT is given by − 

Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)    n = 0,1,….,N−1  …eq(4)

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

Twiddle Factor

It is denoted as WN and defined as  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)  Its magnitude is always maintained at unity. Phase of  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE). It is a vector on unit circle and is used for computational convenience. Mathematically, it can be shown as −

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

  • It is function of r and period N.

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

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

Linear Transformation

Let us understand Linear Transformation −

We know that,

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

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

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

  • Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE) N point vector of signal xN
  • Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE) N point vector of signal XN
    Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

    N - point DFT in matrix term is given by - XN = WNxN

Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)  Matrix of linear transformation

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

IDFT in Matrix form is given by

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

Comparing both the expressions of Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE) and   Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

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

From periodic property of WN and from its symmetric property, it can be concluded that, Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

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. xp(n) of period N. and  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE) . . 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,

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

The new finite sequence can be represented as

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

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.

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

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

Circularly even sequence − Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

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

Conjugate even −  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

Circularly odd sequence −  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
Conjugate odd −  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

Now,  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

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

For any real signal x(n),
Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)

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

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

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 Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Circular time shift   Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Circular frequency shift Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Complex conjugate properties
    Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Multiplication of two sequence
    Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Circular convolution − and multiplication of two DFT
    Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)    m = 0,1,2,....,N - 1
    Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Circular correlation −  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE) , then there exists a cross correlation sequence denoted as Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)  such that  Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
  • Parseval’s TheoremTime Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
    Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE)
The document Time Frequency Transform | Signals and Systems - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Signals and Systems.
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