Continuous-Time Fourier Methods M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1.

Continuous-Time Fourier Methods

M. J. Roberts - All Rights Reserved. Edited

by Dr. Robert Akl

1

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.


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Real and Complex Sinusoids


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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.)


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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.


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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.


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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.


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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.


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Continuous-Time Fourier Series

Definition


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Orthogonality


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Orthogonality


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Orthogonality


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Orthogonality


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Continuous-Time Fourier Series

Definition


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CTFS of a Real Function


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The Trigonometric CTFS


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The Trigonometric CTFS


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CTFS Example #1


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CTFS Example #1


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CTFS Example #2


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CTFS Example #2


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CTFS Example #3


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CTFS Example #3


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CTFS Example #3


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CTFS Example #3


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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 #4


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CTFS Example #4


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CTFS Example #4


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CTFS Example #4


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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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The Sinc Function


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CTFS Example #5


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CTFS Example #5


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CTFS Example #5


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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.


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CTFS of Even and Odd Functions


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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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For discontinuous signals, convergence is exact at every point of continuity.

Discontinuous Signal

Partial CTFS Sums


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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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Samples from x(t)


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CTFS Properties

Linearity


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CTFS Properties

Time Shifting


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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 Properties


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CTFS PropertiesTime Scaling (continued)


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CTFS Properties


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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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CTFS Properties


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CTFS Properties


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CTFS Properties


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Some Common CTFS Pairs


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CTFS Examples


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CTFS Examples


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CTFS Examples


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LTI Systems with Periodic Excitation


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Continuous-Time Fourier Methods

M. J. Roberts - All Rights Reserved. Edited

by Dr. Robert Akl

67

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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CTFS-to-CTFT Transition


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Definition of the CTFT


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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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Some CTFT Pairs


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Convergence and the Generalized Fourier

Transform


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Transform


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Transform


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Transform


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Transform


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Negative FrequencyThis signal is obviously a sinusoid. How is it describedmathematically?


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Negative Frequency


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Negative Frequency


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More CTFT Pairs


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CTFT Properties

Linearity


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CTFT Properties


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CTFT Properties


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CTFT Properties


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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


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CTFT Properties


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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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CTFT Properties


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CTFT Properties


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CTFT Properties


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CTFT Properties


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CTFT Properties


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CTFT Properties


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Numerical Computation of the CTFT


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Continuous-Time Fourier Methods M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1.

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