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

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2Fundamental of Communication Systems ELCT332 Fall2011 Figure 3.1 Construction of a periodic signal by periodic extension of g(t). Aperiodic Signal: Fourier Integral

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3Fundamental of Communication Systems ELCT332 Fall2011 Figure 3.2 Change in the Fourier spectrum when the period T 0 in Fig. 3.1 is doubled.

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4Fundamental of Communication Systems ELCT332 Fall2011 The Fourier series becomes the Fourier integral in the limit as T 0 →∞.

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5Fundamental of Communication Systems ELCT332 Fall2011 (a) e −at u(t) and (b) its Fourier spectra. Fourier integral G(f): direct Fourier transform of g(t) g(t): inverse Fourier transform of G(f) Find the Fourier transform of Dirichlet Condition Linearity of the Fourier Transform (Superposition Theorem)

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6Fundamental of Communication Systems ELCT332 Fall2011 Analogy for Fourier transform. 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.

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7Fundamental of Communication Systems ELCT332 Fall2011 Time-limited pulse. G(f): Spectrum of g(t)

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8Fundamental of Communication Systems ELCT332 Fall2011 Rectangular pulse. Unit Rectangular Function Transforms of some useful functions

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9Fundamental of Communication Systems ELCT332 Fall2011 Triangular pulse. Unit Triangular Function

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10Fundamental of Communication Systems ELCT332 Fall2011 Sinc pulse. Sinc Function

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11Fundamental of Communication Systems ELCT332 Fall2011 (a) Rectangular pulse and (b) its Fourier spectrum. Example

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12Fundamental of Communication Systems ELCT332 Fall2011 (a) Unit impulse and (b) its Fourier spectrum. Example II

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13Fundamental of Communication Systems ELCT332 Fall2011 (a) Constant (dc) signal and (b) its Fourier spectrum. Example III

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14Fundamental of Communication Systems ELCT332 Fall2011 (a) Cosine signal and (b) its Fourier spectrum. Find the inverse Fourier transform of

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15Fundamental of Communication Systems ELCT332 Fall2011 Sign function.

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16Fundamental of Communication Systems ELCT332 Fall2011 Near symmetry between direct and inverse Fourier transforms. Time-Frequency Duality Dual Property

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17Fundamental of Communication Systems ELCT332 Fall2011 Duality property of the Fourier transform. Dual Property

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18Fundamental of Communication Systems ELCT332 Fall2011 The scaling property of the Fourier transform. Time-Scaling Property Time compression of a signal results in spectral expansion, and time expansion of the signal results in its spectral compression.

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19Fundamental of Communication Systems ELCT332 Fall2011 (a) e −a|t| and (b) its Fourier spectrum. Example Prove that and if to find the Fourier transforms of and

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20Fundamental of Communication Systems ELCT332 Fall2011 Physical explanation of the time-shifting property. Time-Shifting Property Delaying a signal by t 0 seconds does not change its amplitude spectrum. The phase spectrum is changed by -2πft 0. To achieve the same time delay, higher frequency sinusoids must undergo proportionately larger phase shifts. Question: Prove that

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21Fundamental of Communication Systems ELCT332 Fall2011 Effect of time shifting on the Fourier spectrum of a signal. Example Find the Fourier transform of Linear phase spectrum

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22Fundamental of Communication Systems ELCT332 Fall2011 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=f 0 Amplitude Modulation Carrier, Modulating signal, Modulated signal

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23Fundamental of Communication Systems ELCT332 Fall2011 Example of spectral shifting by amplitude modulation. Example: Find the Fourier transform of the modulated signal g(t)cos2πf 0 t in which g(t) is a rectangular pulse Frequency division multiplexing (FDM)

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24Fundamental of Communication Systems ELCT332 Fall2011 (a) Bandpass signal and (b) its spectrum. Bandpass Signals

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25Fundamental of Communication Systems ELCT332 Fall2011 (a) Impulse train and (b) its spectrum. Example: Find the Fourier transform of a general periodic signal g(t) of period T 0

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26Fundamental of Communication Systems ELCT332 Fall2011 Using the time differentiation property to find the Fourier transform of a piecewise-linear signal. Time Differentiation Time Integration Find the Fourier transform of the triangular pulse

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27Fundamental of Communication Systems ELCT332 Fall2011 Properties of Fourier Transform Operations Operationg(t)G(f) Superpositiong1(t)+g2(t)G1(f)+G2(f) Scalar multiplicationkg(t)kG(f) DualityG(t)g(-f) Time scalingg(at) Time shiftingg(t-t0) Frequency ShiftG(f-f0) Time convolutiong1(t)*g2(t)G1(f)G2(f) Frequency convolutiong1(t)g2(t)G1(f)*G2(f) Time differentiation Time integration

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28Fundamental of Communication Systems ELCT332 Fall2011 Signal transmission through a linear time-invariant system. H(f): Transfer function/frequency response Signal Transmission Through a Linear System

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29Fundamental of Communication Systems ELCT332 Fall2011 Linear time invariant system frequency response for distortionless transmission. Distortionless transmission: a signal to pass without distortion delayed ouput retains the waveform

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30Fundamental of Communication Systems ELCT332 Fall2011 (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?

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31Fundamental of Communication Systems ELCT332 Fall2011 (a) Ideal low-pass filter frequency response and (b) its impulse response. Ideal filters: allow distortionless transmission of a certain band of frequencies and suppress all the remaining frequencies.

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32Fundamental of Communication Systems ELCT332 Fall2011 Ideal high-pass and bandpass filter frequency responses. Paley-Wiener criterion

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33Fundamental of Communication Systems ELCT332 Fall2011 Approximate realization of an ideal low-pass filter by truncating its impulse response. For a physically realizable system h(t) must be causal h(t)=0 for t<0

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34Fundamental of Communication Systems ELCT332 Fall2011 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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35Fundamental of Communication Systems ELCT332 Fall2011 Basic diagram of a digital filter in practical applications. Digital Filters Sampling, quantizing, and coding

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36Fundamental of Communication Systems ELCT332 Fall2011 Pulse is dispersed when it passes through a system that is not distortionless. Linear Distortion Magnitude distortion Phase Distortion: Spreading/dispersion

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37Fundamental of Communication Systems ELCT332 Fall2011 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. Distortion Caused by Channel Nonlinearities

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38Fundamental of Communication Systems ELCT332 Fall2011 Multipath transmission. Multipath Effects

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39Fundamental of Communication Systems ELCT332 Fall2011 Interpretation of the energy spectral density of a signal. Signal Energy: Parseval’s Theorem Energy Spectral Density

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40Fundamental of Communication Systems ELCT332 Fall2011 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.

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41Fundamental of Communication Systems ELCT332 Fall2011 Find the essential bandwidth where it contains at least 90% of the pulse energy.

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42Fundamental of Communication Systems ELCT332 Fall2011 Energy spectral densities of (a) modulating and (b) modulated signals. Energy of Modulated Signals

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43Fundamental of Communication Systems ELCT332 Fall2011 Figure 3.42 Computation of the time autocorrelation function. Autocorrelation Function Determine the ESD of

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44Fundamental of Communication Systems ELCT332 Fall2011 Limiting process in derivation of PSD. Signal Power Power Spectral Density Time Autocorrelation Function of Power Signals PSD of Modulated Signals

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