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

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

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

Figure 3.2 Change in the Fourier spectrum when the period T0 in Fig. 3.1 is doubled.
Fundamental of Communication Systems ELCT Fall2011

The Fourier series becomes the Fourier integral in the limit as T0 →∞.
Fundamental of Communication Systems ELCT Fall2011

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

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

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

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

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

Sinc Function Sinc pulse.
Fundamental of Communication Systems ELCT Fall2011

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

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

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

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

Sign function. Fundamental of Communication Systems ELCT Fall2011

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

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

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

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

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

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

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

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

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

(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

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

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

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

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

(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

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

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

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

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

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

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

(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

Multipath transmission.
Multipath Effects Multipath transmission. Fundamental of Communication Systems ELCT Fall2011

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

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

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

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

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

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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