Post on 13-Jan-2016
Chapter 2
Signals and Spectra
(All sections, except Section 8, are covered.)
Physically Realizable Waveform
1. Non zero over finite duration (finite energy)
2. Non zero over finite frequency range (physical limitation of media)
3. Continuous in time (finite bandwidth)4. Finite peak value (physical limitation of
equipment)5. Real valued (must be observable)
• Power Signal: finite power, infinite energy
• Energy Signal: finite energy, non-zero power over limited time
• All physical signals are energy signals. Nothing can have infinite power. However, mathematically it is more convenient to deal with power signals. We will use power signals to approximate the behavior of energy signals over the time intervals of interest.
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Signal is a measurable, physical quantity which carries information. In time, it is quantified as w(t). Sometimes it is convenient to view through its frequency components.
Fourier Transform (FT) is a mathematical tool to identify the presence of frequency component for any wave form.
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Properties of Fourier Transform
1. If w(t) is real, W(-f) = W*(f).
2. Linearity:
a1w1(t)+a2w2(t) a1W1(f) + a2 W2(f)
3. Time delay: w(t – T) = W(f) e-j2fT
4. Frequency Translation:
w(t) ej2fot W(f – fo)
5. Convolution: w1(t) w2(t) W1(f)W2(f)
6. Multiplication: w1(t)w2(t) W1(f) W2(f)Note:* is complex conjugate. is convolution integral.X
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-
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Spectrum of Sine Wave
Figure 2–8 Waveform and spectrum of a switched sinusoid.Spectrum of Truncated Sine Wave
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Supposed t is fixed at an arbitrary value.Within the integration, w2(t-) is a “horizontally flipped about=0, and move to the right by t version” of w2(). Now, multiply w1() with w2(t-) for each point of .Then, integrate over - < < . The result is w3(t) for this fixed value of t.Repeat this process for all values of t, - < t < .
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Note: Sections 2.7 and 2.9 will be covered briefly. Section 2.8 will not be covered.
functions. Sinc of sumcertain a toequivalent are signals dBandlimite :tionInterpreta
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Dimensionality Theorem
For a bandlimited waveform with bandwidth B hertz, if the waveform can be completely specified (i.e., later reconstructed by an ideal low pass filter) by N=2BTo samples during a time period of To,then N is the dimension of the wave form.
Conversely, to estimate the bandwidth of a waveform, find a numberN such that N=2BTo is the minimum number of samples needed to reconstructthe waveform during a time period To. Then B follows.
As To , any approximation goes to zero.
A slightly modified version of this theorem is the Bandpass Dimensionality Theorem: Any bandpass waveform (with bandwidth B) can be determined by N=2BTo samples taken during a period of To.
Data Rate Theorem (Corollary to Dimensionality Theorem)
The maximum number of independent quantities which can be transmitted by a bandlimited channel (B hertz) during a time period of To is N=2BTo.
Definition. The baud rate of a digital communication system is the rate of symbols or quantities transmitted per second.
From the Data Rate Theorem, the maximum baud rate of a system with a bandlimited channel (B hertz) is 2B symbols / second.
Definition. The data rate (or bit rate), R, of a system is the baud rate times the information content per symbol (H): R= 2BH bits / second
Suppose a source transmits one of M equally likely symbols. The information
content of each symbol: H = log 2 (1/probablity of each symbol) = log2 M R= 2Blog2 M Data rate is (also known as the Channel Capacity) is determined by (1) channelbandwidth and (2) channel SNR.
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