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Figure 3.1 Construction of a periodic signal by periodic extension of g(t).
Aperiodic Signal: Fourier Integral
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Figure 3.2 Change in the Fourier spectrum when the period T0 in Fig. 3.1 is doubled.
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The Fourier series becomes the Fourier integral in the limit as T0 →∞.
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(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)
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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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Rectangular pulse.
Unit Rectangular Function
Transforms of some useful functions
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(a) Rectangular pulse and (b) its Fourier spectrum.
Example
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(a) Unit impulse and (b) its Fourier spectrum.
Example II
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(a) Constant (dc) signal and (b) its Fourier spectrum.
Example III
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(a) Cosine signal and (b) its Fourier spectrum.
Find the inverse Fourier transform of
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Near symmetry between direct and inverse Fourier transforms.
Time-Frequency Duality
Dual Property
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Duality property of the Fourier transform.
Dual Property
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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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(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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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
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Effect of time shifting on the Fourier spectrum of a signal.
Example
Find the Fourier transform of
Linear phase spectrum
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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 ModulationCarrier, Modulating signal, Modulated signal
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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)
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(a) Bandpass signal and (b) its spectrum.
Bandpass Signals
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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
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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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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
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Signal transmission through a linear time-invariant system.
H(f): Transfer function/frequency response
Signal Transmission Through a Linear System
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Linear time invariant system frequency response for distortionless transmission.
Distortionless transmission: a signal to pass without distortiondelayed ouput retains the waveform
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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?
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(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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Ideal high-pass and bandpass filter frequency responses.
Paley-Wiener criterion
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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
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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
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Basic diagram of a digital filter in practical applications.
Digital Filters
Sampling, quantizing, and coding
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Pulse is dispersed when it passes through a system that is not distortionless.
Linear Distortion
Magnitude distortionPhase Distortion: Spreading/dispersion
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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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Interpretation of the energy spectral density of a signal.
Signal Energy: Parseval’s Theorem
Energy Spectral Density
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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.
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Find the essential bandwidth where it contains at least 90% of the pulse energy.
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Energy spectral densities of (a) modulating and (b) modulated signals.
Energy of Modulated Signals
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Figure 3.42 Computation of the time autocorrelation function.
Autocorrelation FunctionDetermine the ESD of