1 c h a p t e r 3 analysis and transmission of signals

1

C H A P T E R 3

ANALYSIS AND TRANSMISSION OF SIGNALS

2Fundamental of Communication Systems ELCT332 Fall2011

Figure 3.1 Construction of a periodic signal by periodic extension of g(t).

Aperiodic Signal: Fourier Integral


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


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


(a) e−atu(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)


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.


Time-limited pulse.

G(f): Spectrum of g(t)


Rectangular pulse.

Unit Rectangular Function

Transforms of some useful functions


Triangular pulse.

Unit Triangular Function


Sinc pulse.

Sinc Function


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

Example


(a) Unit impulse and (b) its Fourier spectrum.

Example II


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

Example III


(a) Cosine signal and (b) its Fourier spectrum.

Find the inverse Fourier transform of


Sign function.


Near symmetry between direct and inverse Fourier transforms.

Time-Frequency Duality

Dual Property


Duality property of the Fourier transform.

Dual Property


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.


(a) e−a|t| and (b) its Fourier spectrum.

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


Physical explanation of the time-shifting property.

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


Effect of time shifting on the Fourier spectrum of a signal.

Example

Find the Fourier transform of

Linear phase spectrum


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 ModulationCarrier, Modulating signal, Modulated signal


Example of spectral shifting by amplitude modulation.

Example:Find the Fourier transform of the modulated signal g(t)cos2πf0t in which g(t) is a rectangular pulse

Frequency division multiplexing (FDM)


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

Bandpass Signals


(a) Impulse train and (b) its spectrum.

Example:Find the Fourier transform of a general periodic signal g(t) of period T0


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


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

|a|G(f/a)/

G(f)e ft0-j2

g(t)e tfj2 0

g)/dt(d nn )()2( fGfj n

tdxxg )( )()0(

2

1

2

)(fG

fj

fG


Signal transmission through a linear time-invariant system.

H(f): Transfer function/frequency response

Signal Transmission Through a Linear System


Linear time invariant system frequency response for distortionless transmission.

Distortionless transmission: a signal to pass without distortiondelayed ouput retains the waveform


(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) 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.


Ideal high-pass and bandpass filter frequency responses.

Paley-Wiener criterion


Approximate realization of an ideal low-pass filter by truncating its impulse response.

For a physically realizable system h(t) must be causalh(t)=0 for t<0


Butterworth filter characteristics.

The half-power bandwidth•The bandwidth over which the amplitude response remains constant within 3dB.•cut-off frequency


Basic diagram of a digital filter in practical applications.

Digital Filters

Sampling, quantizing, and coding


Pulse is dispersed when it passes through a system that is not distortionless.

Linear Distortion

Magnitude distortionPhase Distortion: Spreading/dispersion


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


Multipath transmission.

Multipath Effects


Interpretation of the energy spectral density of a signal.

Signal Energy: Parseval’s Theorem

Energy Spectral Density


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.


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


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

Energy of Modulated Signals


Figure 3.42 Computation of the time autocorrelation function.

Autocorrelation FunctionDetermine the ESD of


Limiting process in derivation of PSD.

Signal Power

Power Spectral Density

Time Autocorrelation Function of Power Signals

PSD of Modulated Signals

1 c h a p t e r 3 analysis and transmission of signals

Documents

fourier spectrum

fourier integral slide

direct fourier

fourier series

fourier representation

fourier spectra

spectrum of gt slide

fourier integral gf