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
Earlier, we studied sampling in frequency domain. With that basic knowledge, we sample X(ejω) 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 −
…eq(3)
And IDFT is given by −
n = 0,1,….,N−1 …eq(4)
Twiddle Factor
It is denoted as WN 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.
Matrix of linear transformation
IDFT in Matrix form is given by
Comparing both the expressions of and
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,
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 . . 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 0th 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)
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