Lecture 2 Signals and Systems (II)

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Lecture 2Signals and Systems (II)

• Principles of Communications
• Fall 2008
• NCTU EE Tzu-Hsien Sang

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Outlines

• Signal Models Classifications
• Signal Space Orthogonal Basis
• Fourier Series Transform
• Power Spectral Density Correlation
• Signals Linear Systems
• Sampling Theory
• DFT FFT

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More on LTI Systems

• A system is BIBO if output is bounded, given any
  bounded input.
• A system is causal if current output does not
  depend on future input or current input does not
  contribute to the output in the past.

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• Paley-Wiener Condition
• Remarks (1) H(f) cannot grow too fast.
• (2) H(f) cannot be exactly zero over a finite
  band of frequency.
• 2nd ver.

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Eigenfunctions of LTI Systems

• Another way of taking complicated things part.
• Instead of trying to find a set of orthogonal
  basis functions, lets look for signals that will
  not be changed fundamentally when passing them
  through an LTI system.
• Why?
• Consider the key words analysis/synthesis.
• Note Eigen-analysis is not necessarily
  consistent with orthogonal basis analysis.

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• If , where a is a
  constant, then a is the eigenvalue for the
  eigenfunction g(t).
• Let

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• (Cross)correlation functions related by LTI
  systems
• Note In proving them, we use

Filters!!!
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• Since almost any input x(t) can be represented by
  a linear combination of orthogonal sinusoidal
  basis functions , we only need to input
  to the system to characterize the
  systems properties, and the eigenvalue
• carries all the system information responding to
  . (Frequency response!!!)
• In communications, signal distortion is of
  primary concern in high-quality transmission of
  data. Hence, the transmission channel is the key
  investigation target.

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• Three major types of distortion caused by a
  transmission channel
• 1. Amplitude distortion linear system but the
  amplitude response is not constant.
• 2. Phase (delay) distortion linear system but
  the phase shift is not a linear function of
  frequency. (Q What good is linear phase?)
• 3. Nonlinear distortion nonlinear system

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• Example Group Delay

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• Example Ideal general filters

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• Realizable filters approximating ideal filters

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The Uncertainty Principle

• It can be argued that a narrow time signal has a
  wide (frequency) bandwidth, and vice versa

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Sampling Theory

• Youve probably heard of signal processing. But
  how to process a signal?
• For instance, the rectifier maxx(t), 0.
• But, how to do Fourier transform of an arbitrary
  signal x(t)?
• Computers seem a good idea. But computers can
  only work on numbers.
• We need to transform the signal first into
  numbers.
• Q Tell discrete signals from digital signals.

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Hopefully the math becomes easier in ideal case.
The concept actually is harder.

• Ideal sampling signal impulse train (an analog
  signal) , T the sampling
  period
• Analog (continuous-time) signal
• Sampled (continuous-time) signal

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• Aliasing If The replicas of X(f)
  overlap in frequency domain. That is, the higher
  frequency components of overlap with the lower
  frequency components of X(f-fs).

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• Nyquist Sampling Theorem
• Let x(t) be a bandlimited signal with X(f) 0
  for . (i.e., no components at
  frequencies greater than W.) Then x(t) is
  uniquely determined by its samples
  if .

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• In other words, oversampling preserves all the
  information that x(t) contains. It is possible to
  reconstruct x(t) purely by its samples.
• Ideal reconstruction filter (interpretation in
  frequency domain
• In time domain

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• Two types of reconstruction errors

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DFT FFT

• You can view DFT as a totally new definition for
  a totally different set of signals. Or you can
  try to connect it to the Fourier Transform.

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• Fast Fourier Transform (FFT) is not a new
  transform, it is simply a fast way to compute
  DFT. So, dont use FFT to denote the object that
  you want to compute only use it to denote the
  tool that you use to compute it. (Gauss knew the
  method already!)
• Application example Fast convolution via FFT