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

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**Aperiodic Signal: Fourier Integral**

Figure 3.1 Construction of a periodic signal by periodic extension of g(t). Fundamental of Communication Systems ELCT Fall2011

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**Figure 3.2 Change in the Fourier spectrum when the period T0 in Fig. 3.1 is doubled.**

Fundamental of Communication Systems ELCT Fall2011

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**The Fourier series becomes the Fourier integral in the limit as T0 →∞.**

Fundamental of Communication Systems ELCT Fall2011

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**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) Fundamental of Communication Systems ELCT Fall2011

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**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. Fundamental of Communication Systems ELCT Fall2011

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**G(f): Spectrum of g(t) Time-limited pulse.**

Fundamental of Communication Systems ELCT Fall2011

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**Transforms of some useful functions**

Unit Rectangular Function Rectangular pulse. Fundamental of Communication Systems ELCT Fall2011

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**Unit Triangular Function**

Triangular pulse. Fundamental of Communication Systems ELCT Fall2011

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**Sinc Function Sinc pulse.**

Fundamental of Communication Systems ELCT Fall2011

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**Example (a) Rectangular pulse and (b) its Fourier spectrum.**

Fundamental of Communication Systems ELCT Fall2011

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**Example II (a) Unit impulse and (b) its Fourier spectrum.**

Fundamental of Communication Systems ELCT Fall2011

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**Example III (a) Constant (dc) signal and (b) its Fourier spectrum.**

Fundamental of Communication Systems ELCT Fall2011

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**Find the inverse Fourier transform of**

(a) Cosine signal and (b) its Fourier spectrum. Fundamental of Communication Systems ELCT Fall2011

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Sign function. Fundamental of Communication Systems ELCT Fall2011

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**Time-Frequency Duality**

Dual Property Near symmetry between direct and inverse Fourier transforms. Fundamental of Communication Systems ELCT Fall2011

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**Dual Property Duality property of the Fourier transform.**

Fundamental of Communication Systems ELCT Fall2011

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**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. Fundamental of Communication Systems ELCT Fall2011

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**Prove that and if to find the Fourier transforms of and Example**

(a) e−a|t| and (b) its Fourier spectrum. Fundamental of Communication Systems ELCT Fall2011

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**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. Fundamental of Communication Systems ELCT Fall2011

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**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. Fundamental of Communication Systems ELCT Fall2011

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**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. Fundamental of Communication Systems ELCT Fall2011

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**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. Fundamental of Communication Systems ELCT Fall2011

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**(a) Bandpass signal and (b) its spectrum.**

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

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**(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. Fundamental of Communication Systems ELCT Fall2011

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**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. Fundamental of Communication Systems ELCT Fall2011

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**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 Fundamental of Communication Systems ELCT Fall2011

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**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. Fundamental of Communication Systems ELCT Fall2011

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**Distortionless transmission: a signal to pass without distortion **

delayed ouput retains the waveform Linear time invariant system frequency response for distortionless transmission. Fundamental of Communication Systems ELCT Fall2011

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**(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. Fundamental of Communication Systems ELCT Fall2011

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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. Fundamental of Communication Systems ELCT Fall2011

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**Ideal high-pass and bandpass filter frequency responses.**

Paley-Wiener criterion Fundamental of Communication Systems ELCT Fall2011

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**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. Fundamental of Communication Systems ELCT Fall2011

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**Butterworth filter characteristics.**

The half-power bandwidth The bandwidth over which the amplitude response remains constant within 3dB. cut-off frequency Fundamental of Communication Systems ELCT Fall2011

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**Basic diagram of a digital filter in practical applications.**

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

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**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. Fundamental of Communication Systems ELCT Fall2011

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**(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. Fundamental of Communication Systems ELCT Fall2011

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**Multipath transmission.**

Multipath Effects Multipath transmission. Fundamental of Communication Systems ELCT Fall2011

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**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. Fundamental of Communication Systems ELCT Fall2011

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**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. Fundamental of Communication Systems ELCT Fall2011

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**Find the essential bandwidth where it contains at least 90% of the pulse energy.**

Fundamental of Communication Systems ELCT Fall2011

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**Energy spectral densities of (a) modulating and (b) modulated signals.**

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

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**Figure 3.42 Computation of the time autocorrelation function.**

Determine the ESD of Figure 3.42 Computation of the time autocorrelation function. Fundamental of Communication Systems ELCT Fall2011

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**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 Fundamental of Communication Systems ELCT Fall2011

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