The document discusses Fourier transforms of discrete signals. It covers topics like sampling, the discrete time Fourier transform (DTFT), the discrete Fourier transform (DFT), and fast Fourier transform (FFT) algorithms. The key points are: - Continuous signals are converted to discrete signals through sampling at discrete time intervals. The sampling rate must be high enough to avoid aliasing. - The DTFT analyzes the frequency spectrum of a discrete signal and is periodic in frequency space. The DFT provides the frequency spectrum of a finite discrete signal. - FFT algorithms like the radix-2 algorithm improve the efficiency of computing DFTs and DFTs by decomposing the computation into smaller subproblems.

1. Fourier Transforms of Discrete
Signals
Mr. HIMANSHU DIWAKAR
Assistant Professor
JETGI
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2. Mr. HIMANSHU DIWAKAR JETGI 2

3. Sampling
• Continuous signals are digitized using digital computers
• When we sample, we calculate the value of the continuous
signal at discrete points
• How fast do we sample
• What is the value of each point
• Quantization determines the value of each samples value
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4. Sampling Periodic Functions
- Note that wb = Bandwidth, thus if then aliasing occurs (signal overlaps)
-To avoid aliasing
-According sampling theory:
To hear music up to 20KHz a CD
should sample at the rate of 44.1 KHzMr. HIMANSHU DIWAKAR JETGI 4

5. Discrete Time Fourier Transform
• In likely we only have access to finite amount of data sequences (after
sampling)
• Recall for continuous time Fourier transform, when the signal is
sampled:
• Assuming
• Discrete-Time Fourier Transform (DTFT):
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6. Discrete Time Fourier Transform
• Discrete-Time Fourier Transform (DTFT):
• A few points
• DTFT is periodic in frequency with period of 2p
• X[n] is a discrete signal
• DTFT allows us to find the spectrum of the discrete signal as viewed from a
window
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7. Example of Convolution
• Convolution
• We can write x[n] (a periodic function) as an infinite sum of the function xo[n]
(a non-periodic function) shifted N units at a time
• This will result
• Thus
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8. Finding DTFT For periodic signals
• Starting with xo[n]
• DTFT of xo[n]
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9. Example A & B
X[n]=a|n|, 0 < a < 1.
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10. Table of Common Discrete Time Fourier Transform (DTFT) Pairs
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11. Table of Common Discrete Time Fourier Transform (DTFT) Pairs
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12. Discrete Fourier Transform
• We often do not have an infinite amount of data which is required by
DTFT
• For example in a computer we cannot calculate uncountable infinite
(continuum) of frequencies as required by DTFT
• Thus, we use DTF to look at finite segment of data
• We only observe the data through a window
• In this case the xo[n] is just a sampled data between n=0, n=N-1 (N points)
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13. Discrete Fourier Transform
• It turns out that DFT can be defined as
• Note that in this case the points are spaced 2pi/N; thus the resolution of the
samples of the frequency spectrum is 2pi/N.
• We can think of DFT as one period of discrete Fourier series
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14. A short hand notation
Remember:
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15. Inverse of DFT
• We can obtain the inverse of DFT
• Note that
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16. Example of DFT
• Find X[k]
• We know k=1,.., 7; N=8
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17. Example of DFT
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18. Example of DFT
Time shift Property of DFT
Polar plot for
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19. Example of DFT
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20. Example of DFT
Summation for X[k]
Using the shift property!
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21. Example of IDFT
Remember:
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22. Example of IDFT
Remember:
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23. Fast Fourier Transform Algorithms
• Consider DTFT
• Basic idea is to split the sum into 2 subsequences of length N/2 and continue all
the way down until you have N/2 subsequences of length 2
Log2(8)
N
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24. Radix-2 FFT Algorithms - Two point FFT
• We assume N=2^m
• This is called Radix-2 FFT Algorithms
• Let’s take a simple example where only two points are given n=0, n=1; N=2
y0
y1
Butterfly FFT
Advantage: Less
computationally
intensive: N/2.log(N)
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25. General FFT Algorithm
• First break x[n] into even and odd
• Let n=2m for even and n=2m+1 for odd
• Even and odd parts are both DFT of a N/2
point sequence
• Break up the size N/2 subsequent in half by
letting 2mm
• The first subsequence here is the term x[0],
x[4], …
• The second subsequent is x[2], x[6], …
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26. Example
1
1)2sin()2cos(
)]12[(]2[][
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2
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Let’s take a simple example where only two points are given n=0, n=1; N=2
]1[]0[]1[]0[)]1[(]0[]1[
]1[]0[)]1[(]0[]0[
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mm
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Same result
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27. FFT Algorithms - Four point FFT
First find even and odd parts and then combine them:
The general form:
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28. FFT Algorithms - 8 point FFT
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29. THANK YOU
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