1. C H A P T E R 3
ANALYSIS AND TRANSMISSION
OF SIGNALS

2. Aperiodic Signal: Fourier Integral
Figure 3.1 Construction of a periodic signal by periodic extension of g(t).
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3. Figure 3.2 Change in the Fourier spectrum when the period T0 in Fig. 3.1 is doubled.
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4. The Fourier series becomes the Fourier integral in the limit as T0 →∞.
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5. G(f): direct Fourier transform of g(t)
Fourier integral
G(f): direct Fourier transform of g(t)
g(t): inverse Fourier transform of G(f)
Find the Fourier transform of
(a) e−atu(t) and (b) its Fourier spectra.
Dirichlet Condition
Linearity of the Fourier Transform (Superposition Theorem)
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6. Physical Appreciation of the Fourier Transform
Fourier representation is a way of a signal in terms of everlasting sinusoids, or exponentials.
The Fourier Spectrum of a signal indicates the relative amplitudes and phases of the sinusoids that are required to synthesize the signal.
Analogy for Fourier transform.
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7. G(f): Spectrum of g(t) Time-limited pulse.
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8. Transforms of some useful functions
Unit Rectangular Function
Rectangular pulse.
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9. Unit Triangular Function
Triangular pulse.
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10. Sinc Function Sinc pulse.
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11. Example (a) Rectangular pulse and (b) its Fourier spectrum.
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12. Example II (a) Unit impulse and (b) its Fourier spectrum.
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13. Example III (a) Constant (dc) signal and (b) its Fourier spectrum.
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14. Find the inverse Fourier transform of
(a) Cosine signal and (b) its Fourier spectrum.
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15. Sign function.
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16. Time-Frequency Duality
Dual Property
Near symmetry between direct and inverse Fourier transforms.
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17. Dual Property Duality property of the Fourier transform.
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18. Time-Scaling Property
Time compression of a signal results in spectral expansion, and time expansion of the signal results in its spectral compression.
The scaling property of the Fourier transform.
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19. Prove that and if to find the Fourier transforms of and Example
(a) e−a|t| and (b) its Fourier spectrum.
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20. Physical explanation of the time-shifting property.
Delaying a signal by t0 seconds does not change its amplitude spectrum. The phase spectrum is changed by -2πft0 .
To achieve the same time delay, higher frequency sinusoids must undergo proportionately larger phase shifts.
Question: Prove that
Physical explanation of the time-shifting property.
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21. Effect of time shifting on the Fourier spectrum of a signal.
Example
Find the Fourier transform of
Linear phase spectrum
Effect of time shifting on the Fourier spectrum of a signal.
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22. Amplitude modulation of a signal causes spectral shifting.
Frequency-Shifting Property
Multiplication of a signal by a factor shifts the spectrum of that signal by f=f0
Amplitude Modulation
Carrier, Modulating signal, Modulated signal
Amplitude modulation of a signal causes spectral shifting.
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23. Frequency division multiplexing (FDM)
Example:
Find the Fourier transform of the modulated signal g(t)cos2πf0t in which g(t) is a rectangular pulse
Frequency division multiplexing (FDM)
Example of spectral shifting by amplitude modulation.
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24. (a) Bandpass signal and (b) its spectrum.
Bandpass Signals
(a) Bandpass signal and (b) its spectrum.
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25. (a) Impulse train and (b) its spectrum.
Example:
Find the Fourier transform of a general periodic signal g(t) of period T0
(a) Impulse train and (b) its spectrum.
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26. Time Differentiation Time Integration
Find the Fourier transform of the triangular pulse
Time Integration
Using the time differentiation property to find the Fourier transform of a piecewise-linear signal.
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27. Properties of Fourier Transform Operations Operation g(t) G(f)
Superposition
g1(t)+g2(t)
G1(f)+G2(f)
Scalar multiplication
kg(t)
kG(f)
Duality
G(t)
g(-f)
Time scaling
g(at)
Time shifting
g(t-t0)
Frequency Shift
G(f-f0)
Time convolution
g1(t)*g2(t)
G1(f)G2(f)
Frequency convolution
g1(t)g2(t)
G1(f)*G2(f)
Time differentiation
Time integration
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28. Signal transmission through a linear time-invariant system.
Signal Transmission Through a Linear System
H(f): Transfer function/frequency response
Signal transmission through a linear time-invariant system.
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29. Distortionless transmission: a signal to pass without distortion
delayed ouput retains the waveform
Linear time invariant system frequency response for distortionless transmission.
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30. (a) Simple RC filter. (b) Its frequency response and time delay.
Determine the transfer function H(f), and td(f).
What is the requirement on the bandwidth of g(t) if amplitude variation within 2% and time delay variation within 5% are tolerable?
(a) Simple RC filter. (b) Its frequency response and time delay.
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31. Ideal filters: allow distortionless transmission of a certain band of frequencies and suppress all the remaining frequencies.
(a) Ideal low-pass filter frequency response and (b) its impulse response.
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32. Ideal high-pass and bandpass filter frequency responses.
Paley-Wiener criterion
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33. For a physically realizable system h(t) must be causal
h(t)= for t<0
Approximate realization of an ideal low-pass filter by truncating its impulse response.
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34. Butterworth filter characteristics.
The half-power bandwidth
The bandwidth over which the amplitude response remains constant within 3dB.
cut-off frequency
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35. Basic diagram of a digital filter in practical applications.
Digital Filters
Sampling, quantizing, and coding
Basic diagram of a digital filter in practical applications.
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36. Phase Distortion: Spreading/dispersion
Linear Distortion
Magnitude distortion
Phase Distortion: Spreading/dispersion
Pulse is dispersed when it passes through a system that is not distortionless.
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37. (d) spectrum of the received signal after low-pass filtering.
Distortion Caused by Channel Nonlinearities
Signal distortion caused by nonlinear operation: (a) desired (input) signal spectrum;
(b) spectrum of the unwanted signal (distortion) in the received signal; (c) spectrum of the received signal;
(d) spectrum of the received signal after low-pass filtering.
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38. Multipath transmission.
Multipath Effects
Multipath transmission.
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39. Interpretation of the energy spectral density of a signal.
Signal Energy: Parseval’s Theorem
Energy Spectral Density
Interpretation of the energy spectral density of a signal.
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40. Figure 3.39 Estimating the essential bandwidth of a signal.
Essential Bandwidth: the energy content of the components of frequeicies greater than B Hz is negligible.
Figure 3.39 Estimating the essential bandwidth of a signal.
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41. Find the essential bandwidth where it contains at least 90% of the pulse energy.
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42. Energy spectral densities of (a) modulating and (b) modulated signals.
Energy of Modulated Signals
Energy spectral densities of (a) modulating and (b) modulated signals.
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43. Figure 3.42 Computation of the time autocorrelation function.
Determine the ESD of
Figure 3.42 Computation of the time autocorrelation function.
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44. Limiting process in derivation of PSD.
Signal Power
Power Spectral Density
Limiting process in derivation of PSD.
Time Autocorrelation Function of Power Signals
PSD of Modulated Signals
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