Electrical Communications Systems 0909.331.01 Spring 2005

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Electrical Communications Systems0909.331.01
Spring 2005
Lecture 3aFebruary 15, 2005

• Shreekanth Mandayam
• ECE Department
• Rowan University
• http//engineering.rowan.edu/shreek/spring05/ecom
  ms/

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Plan

• CFTs for periodic waveforms
• Sampling
• Time-limited and Band-limited waveforms
• Nyquist Sampling
• Impulse Sampling
• Dimensionality Theorem
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Relation between CFT and DFT

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ECOMMS Topics
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CFT for Periodic Signals
Recall

• We want to get the CFT for a periodic signal
• What is ?

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CFT for Periodic Signals

• Sine Wave
• w(t) A sin (2pf0t)

• Square Wave

Instrument Demo
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Sampling

• Time-limited waveform
• w(t) 0 t gt T

• Band-limited waveform
• W(f)F(w(t)0 f gt B

• Can a waveform be both time-limited and
  band-limited?

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Nyquist Sampling Theorem

• Any physical waveform can be represented by
• where
• If w(t) is band-limited to B Hz and

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What does this mean?

• If then we can reconstruct w(t)
  without error by summing weighted, delayed sinc
  pulses
• weight w(n/fs)
• delay n/fs
• We need to store only
  samples of w(t),
  i.e., w(n/fs)
• The sinc pulses
  can be generated
  
  as needed (How?)

Matlab Demo sampling.m
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Impulse Sampling

• How do we mathematically represent a sampled
  waveform in the
• Time Domain?
• Frequency Domain?

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Sampling Spectral Effect
Original
Sampled
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Spectral Effect of Sampling
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Aliasing

• If fs lt 2B, the waveform is undersampled
• aliasing or spectral folding
• How can we avoid aliasing?
• Increase fs
• Pre-filter the signal so that it is bandlimited
  to 2B lt fs

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Dimensionality Theorem

• A real waveform can be completely specified by
  
• N 2BT0
  independent pieces of
  information over a time interval T0
• N Dimension of the waveform
• B Bandwidth
• BT0 Time-Bandwidth Product
• Memory calculation for storing the waveform
• fs gt 2B
• At least N numbers must be stored over the time
  interval T0 n/fs

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Discrete Fourier Transform (DFT)

• Discrete Domains
• Discrete Time k 0, 1, 2, 3, , N-1
• Discrete Frequency n 0, 1, 2, 3, , N-1
• Discrete Fourier Transform
• Inverse DFT

Equal frequency intervals
n 0, 1, 2,.., N-1
k 0, 1, 2,.., N-1
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Importance of the DFT

• Allows time domain / spectral domain
  transformations using discrete arithmetic
  operations
• Computational Complexity
• Raw DFT N2 complex operations ( 2N2 real
  operations)
• Fast Fourier Transform (FFT) N log2 N real
  operations
• Fast Fourier Transform (FFT)
• Cooley and Tukey (1965), Butterfly Algorithm,
  exploits the periodicity and symmetry of
  e-j2pkn/N
• VLSI implementations FFT chips
• Modern DSP

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How to get the frequency axis in the DFT

• The DFT operation just converts one set of
  number, xk into another set of numbers Xn -
  there is no explicit definition of time or
  frequency
• How can we relate the DFT to the CFT and obtain
  spectral amplitudes for discrete frequencies?

(N-point FFT)
Need to know fs
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DFT Properties

• DFT is periodic
• Xn XnN Xn2N
• I-DFT is also periodic!
• xk xkN xk2N .
• Where are the low and high frequencies on the
  DFT spectrum?

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Relation between CFT and DFT

• Windowing
• Sampling
• Generation of Periodic Samples

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Summary