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Discrete Fourier Transform, 1-D Discrete Fourier Transform, Fast Fourier Transform, Properities of 2 – D Fourier transform, Walsh, Hadmard, and Discrete cosine transform.
Typology: Study notes
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**1. Aim and objective
II. Pre- Test – MCQ
7.In fourier transform of a real signal the magnitude and phase function will be a) Symmetric and Symmetric b) Symmetric and Anti-Symmetric c) Anti-Symmetric and Symmetric d) Anti- Symmetric and Anti-Symmetric
8.A first order LTI system will behave as a a) low pass filter b) band pass filter c) high pass filter d) low pass or high pass filter
9.The fourier transform of impulse response of a system is called a) transfer function b) frequency response c) forced response d) natural response
a) n – k b) n + k c) 2n d) N^2
Conditions Transform Conditions for Existence of Fourier Transform
Any function f(t) can be represented by using Fourier transform only when the function satisfies Dirichlet’s conditions. i.e.
Discrete Time Fourier Transforms (DTFT)
The discrete-time Fourier transform (DTFT) or the Fourier transform of a discrete–time sequence x[n] is a representation of the sequence in terms of the complex exponential sequence 𝑒𝑖𝜔𝑛
The by The DTFT sequence x[n] is given by
Inverse Transform Inverse Discrete-Time Fourier Transform
The family of Fourier transform
A signal can be either continuous or discrete, and it can be either periodic or aperiodic. The combination of these two features generates the four categories, described below and illustrated in Figure..
Aperiodic-Continuous This includes, for example, decaying exponentials and the Gaussian curve. These signals extend to both positive and negative infinity without repeating in a periodic pattern. The Fourier Transform for this type of signal is simply called the Fourier Transform.
Periodic-Continuous Here the examples include: sine waves, square waves, and any waveform that repeats itself in a regular pattern from negative to positive infinity. This version of the Fourier transform is called the Fourier Series.
Aperiodic-Discrete These signals are only defined at discrete points between positive and negative infinity, and do not repeat themselves in a periodic fashion. This type of Fourier transform is called the Discrete Time Fourier Transform.
Periodic-Discrete These are discrete signals that repeat themselves in a periodic fashion from negative to positive infinity. This class of Fourier Transform is sometimes called the Discrete Fourier Series, but is most often called the Discrete Fourier Transform.
DFT and IDFT
DFT is used for analyzing discrete-time finite-duration signals in the frequency domain. The DFT and IDFT pair of equation is given by
The DFT spectrum is periodic with period N (which is expected, since the DTFT spectrum is periodic as well, but with period 2π).
Example: DFT of a rectangular pulse:
the rectangular pulse is “interpreted” by the DFT as a spectral line at frequency ω= 0.
Zero Padding What happens with the DFT of this rectangular pulse if we increase N by zero padding:
= x(0) + x(1) 𝑒−𝑗𝜋^ + x(2) 𝑒−𝑗2𝜋^ + x(3) 𝑒−𝑗3𝜋 = 1+2(-1)+3(-1)+4(-1)=1-2+3-4=-
For k= 3, 𝑋(3) = ∑^ 𝑋(𝑛)𝑒
−2𝜋(3)𝑛 (^4) 𝑛=0 4
=x(0) + x(1) 𝑒−𝑗3𝜋/2^ + x(2) 𝑒−𝑗3𝜋^ + x(3) 𝑒−𝑗9𝜋/ = 1+2(j) +3(-1)+4(-j) = 1+2j-3-4j = -1-2j
X(K) = {10,-2+10j , -2 , -2-2j}
X(K) = {10, 2.28 ∠135 , 2 , 2.23 ∠ − 116.56 }
Convolution Property of the Fourier Transform
The convolution of two functions in time is defined by:
The Fourier Transform of the convolution of g(t) and h(t) [with corresponding Fourier Transforms G(f) and H(f)] is given by:
Theorem If a discrete-time system linear shift-invariant, T[.], has the unit sample response
T[ δ(n) ] = h(n) then the output y(n) corresponding to any input x(n) is given by
∞
𝑘=−∞
∞
𝑘=−∞ 𝑦(𝑛) = 𝑥(𝑛) ∗ ℎ(𝑛) = ℎ(𝑛) ∗ 𝑥 (𝑛) The second summation is obtained by setting m = n–k ; then for k = – ∞ we have m = + ∞, and for k = ∞ we have m = – ∞. Thus
[ Linear Convolution] Given the input { x(n) } ={1, 2, 3, 1} and the unit sample response { h(n) } = {4, 3, 2, 1} find the response y(n) = x(n) * h(n).
Answer Since x(k) = 0 for k < 0 and h(n – k) = 0 for k > n , the convolution sum becomes
∞
𝑘=−∞
𝑛
𝑘=
Now y(n) can be evaluated for various values of n ; for example, setting n = 0 gives y(0).
Linear Convolution of {x(n)} ={1, 2, 3, 1} and {h(n)} = {4, 3, 2, 1}
𝑛
𝑘=
Tabular Method of Linear convolution
Thus y(n) ={4, 11, 20, 18, 11, 5, 1}
Another method for linear convolution
Step 1 : Multiply X(n) and h(n) in the table
Step 2 : Add the Diagonals
Y(n) = [ 4, 8+3, 12+6+2 , 4+9+4+1, 3+6+2, 2+3, 1 ] = [ 4, 11, 20, 18, 11, 5, 1]
Circular convolution
The convolution property of DFT states that, the multiplication of the DFTs of two sequences is equivalent to the DFT of the circular convolution of the two sequences.
𝑛
𝑘=
Methods of Circular Convolution
Generally, there are two methods, which are adopted to perform circular convolution and they are
Concentric circle method, Matrix multiplication method.
Concentric Circle Method
Let x 1 (n)x1(n) and x 2 (n)x2(n) be two given sequences. The steps followed for circular convolution of x 1 (n)x1(n) and x 2 (n)x2(n) are Take two concentric circles. Plot N samples of x 1 (n)x1(n) on the circumference of the outer circle maintaining equal distance successive points in anti-clockwise direction. For plotting x 2 (n) x2(n), plot N samples of x 2 (n)x2(n) in clockwise direction on the inner circle, starting sample placed at the same point as 0th^ sample of x 1 (n) x1(n) Multiply corresponding samples on the two circles and add them to get output. Rotate the inner circle anti-clockwise with one sample at a time.
Matrix Multiplication Method
Matrix method represents the two given sequence x 1 (n)x1(n) and x 2 (n)x2(n) in matrix form. One of the given sequences is repeated via circular shift of one sample at a time to form a N X N matrix. The other sequence is represented as column matrix. The multiplication of two matrices give the result of circular convolution.
When m=1,
𝑛
𝑘=
When m=2,
𝑛
𝑘=
When m=3,
𝑛
𝑘=
X 3 (n) = {14,16,14,16}
