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Representing a Signal• The convolution method for finding the response of a system to an excitation takes advantage of the linearity and time-invariance of the system and represents the excitation as a linear combination of impulses and the response as a linear combination of impulse responses
• The Fourier series represents a signal as a linear combination of complex sinusoids
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Linearity and Superposition
If an excitation can be expressed as a sum of complex sinusoidsthe response of an LTI system can be expressed as the sum of responses to complex sinusoids.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Jean Baptiste Joseph Fourier
3/21/1768 - 5/16/1830
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Conceptual OverviewThe Fourier series represents a signal as a sum of sinusoids.The best approximation to the dashed-line signal below usingonly a constant is the solidline. (A constant is a cosine of zero frequency.)
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Conceptual OverviewThe best approximation to the dashed-line signal using a constant plus one real sinusoid of the same fundamental frequency as the dashed-line signal is the solid line.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Conceptual OverviewThe best approximation to the dashed-line signal using a constant plus one sinusoid of the same fundamental frequency as the dashed-line signal plus another sinusoid of twice the fundamentalfrequency of the dashed-line signal is the solid line.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Conceptual OverviewThe best approximation to the dashed-line signal using a constant plus three sinusoids is the solid line. In this case (but not in general), the third sinusoid has zero amplitude. This means that no sinusoid of three times the fundamental frequency improves the approximation.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Conceptual OverviewThe best approximation to the dashed-line signal using a constant plus four sinusoids is the solid line. This is a good approximation which gets better with the addition of more sinusoids at higher integer multiples of the fundamental frequency.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Continuous-Time Fourier Series
Definition
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Continuous-Time Fourier Series
Definition
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Linearity of the CTFS
These relations hold only if the harmonic functions of allthe component functions are based on the samerepresentation time T.
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CTFS Example #4A graph of the magnitude and phase of the harmonic functionas a function of harmonic number is a good way of illustrating it.
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CTFS Example #5The CTFS representation of this cosine is the signalbelow, which is an odd function, and the discontinuitiesmake the representation have significant higher harmoniccontent. This is a very inelegant representation.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
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Convergence of the CTFS
For continuous signals, convergence is exact at every point.
A Continuous Signal
Partial CTFS Sums
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Convergence of the CTFS
For discontinuous signals, convergence is exact at every point of continuity.
Discontinuous Signal
Partial CTFS Sums
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Convergence of the CTFS
At points of discontinuitythe Fourier seriesrepresentation convergesto the mid-point of thediscontinuity.
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Numerical Computation of the CTFS
How could we find the CTFS of a signal which has noknown functional description?
Numerically.
Unknown
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Numerical Computation of the CTFS
Samples from x(t)
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CTFS Properties
Frequency Shifting (Harmonic Number
Shifting)
A shift in frequency (harmonic number) corresponds to multiplication of the time function by a complex exponential.
Time Reversal
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CTFS PropertiesTime Scaling (continued)
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CTFS PropertiesChange of Representation Time
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CTFS Properties
Time Differentiation
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CTFS PropertiesTime Integration
is not periodic
Case 1 Case 2
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LTI Systems with Periodic Excitation
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LTI Systems with Periodic Excitation
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LTI Systems with Periodic Excitation
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LTI Systems with Periodic Excitation
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Extending the CTFS
• The CTFS is a good analysis tool for systems with periodic excitation but the CTFS cannot represent an aperiodic signal for all time
• The continuous-time Fourier transform (CTFT) can represent an aperiodic signal for all time
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Some Remarkable Implications of the Fourier Transform
The CTFT expresses a finite-amplitude, real-valued, aperiodic signal which can also, in general, be time-limited, as a summation (an integral) of an infinite continuum of weighted, infinitesimal-amplitude, complex sinusoids, each of which is unlimited intime. (Time limited means “having non-zero values only for a finite time.”)
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Frequency Content
Lowpass Highpass
Bandpass
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Convergence and the Generalized Fourier
Transform
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Convergence and the Generalized Fourier
Transform
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Convergence and the Generalized Fourier
Transform
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Convergence and the Generalized Fourier
Transform
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Convergence and the Generalized Fourier
Transform
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Negative FrequencyThis signal is obviously a sinusoid. How is it describedmathematically?
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The “Uncertainty” PrincipleThe time and frequency scaling properties indicate that if a signal
is expanded in one domain it is compressed in the other domain.This is called the “uncertainty principle” of Fourier analysis.
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CTFT Properties
In the frequency domain, the cascade connection multipliesthe frequency responses instead of convolving the impulseresponses.
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