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Like continuous time signal Fourier transform, discrete time Fourier Transform can be used to represent a discrete sequence into its equivalent frequency domain representation and LTI discrete time system and develop various computational algorithms.
X (jω) in continuous F.T, is a continuous function of x(n). However, DFT deals with representing x(n) with samples of its spectrum X(ω). Hence, this mathematical tool carries much importance computationally in convenient representation. Both, periodic and nonperiodic sequences can be processed through this tool. The periodic sequences need to be sampled by extending the period to infinity.
Frequency Domain Sampling
From the introduction, it is clear that we need to know how to proceed through frequency domain sampling i.e. sampling X(ω). Hence, the relationship between sampled Fourier transform and DFT is established in the following manner.
Similarly, periodic sequences can fit to this tool by extending the period N to infinity.
Let an Non periodic sequence be,
Defining its Fourier transform,
...eq(1)
Here, X(ω) is sampled periodically, at every δω radian interval.
As X(ω) is periodic in 2π radians, we require samples only in fundamental range. The samples are taken after equidistant intervals in the frequency range 0≤ω≤2π. Spacing between equivalent intervals is radian.
Now evaluating,
...eq(2)
where k=0,1,……N1
After subdividing the above, and interchanging the order of summation
...eq(3)
periodic function of period N and its fourier series
where, n = 0,1,…..,N1; ‘p’ stands for periodic entity or function
The Fourier coefficients are,
...eq(4)
Comparing equations 3 and 4, we get ; ...eq(5) ...eq(6)
From Fourier series expansion,
...eq(7)
Where n=0,1,…,N1
Here, we got the periodic signal from X(ω). x(n) can be extracted from x_{p}(n)only, if there is no aliasing in the time domain. N≥L
N = period of x_{p}(n) L = period of x(n)
The mapping is achieved in this manner.
Properties of DFT
Linearity
It states that the DFT of a combination of signals is equal to the sum of DFT of individual signals. Let us take two signals x_{1}(n) and x_{2}(n), whose DFT s are X_{1}(ω) and X_{2}(ω) respectively. So, if
Then
where a and b are constants.
Symmetry
The symmetry properties of DFT can be derived in a similar way as we derived DTFT symmetry properties. We know that DFT of sequence x(n) is denoted by X(K). Now, if x(n) and X(K) are complex valued sequence, then it can be represented as under
And
Duality Property
Let us consider a signal x(n), whose DFT is given as X(K). Let the finite duration sequence be X(N). Then according to duality theorem,
If,
Then,
So, by using this theorem if we know DFT, we can easily find the finite duration sequence.
Complex Conjugate Properties
Suppose, there is a signal x(n), whose DFT is also known to us as X(K). Now, if the complex conjugate of the signal is given as x*(n), then we can easily find the DFT without doing much calculation by using the theorem shown below.
If,
Then,
Circular Frequency Shift
The multiplication of the sequence x(n) with the complex exponential sequence e^{j2Πkn/N} is equivalent to the circular shift of the DFT by L units in frequency. This is the dual to the circular time shifting property.
If,
Then,
Multiplication of Two Sequence
If there are two signal x_{1}(n) and x_{2}(n) and their respective DFTs are X_{1}(k) and X_{2}(K), then multiplication of signals in time sequence corresponds to circular convolution of their DFTs.
If,
Then,
Parseval’s Theorem
For complex valued sequences x(n) and y(n), in general
If,
Then,
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