Continuous-Time Fourier Methods

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

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 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

Linearity and Superposition If an excitation can be expressed as a sum of complex sinusoids the 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

Real and Complex Sinusoids M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Jean Baptiste Joseph Fourier 3/21/1768 - 5/16/1830 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Conceptual Overview The Fourier series represents a signal as a sum of sinusoids. The best approximation to the dashed-line signal below using only a constant is the solid line. (A constant is a cosine of zero frequency.) M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Conceptual Overview The 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

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Conceptual Overview The 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 fundamental frequency of the dashed-line signal is the solid line. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Conceptual Overview The 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

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Conceptual Overview The 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

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

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Orthogonality M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Orthogonality M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Orthogonality M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Orthogonality M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS of a Real Function M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The Trigonometric CTFS M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The Trigonometric CTFS M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #1 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #1 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #2 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #2 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #3 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #3 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #3 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #3 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Linearity of the CTFS These relations hold only if the harmonic functions of all the component functions are based on the same representation time T. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #4 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #4 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #4 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #4 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #4 A graph of the magnitude and phase of the harmonic function as a function of harmonic number is a good way of illustrating it. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl The Sinc Function M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #5 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #5 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #5 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Example #5 The CTFS representation of this cosine is the signal below, which is an odd function, and the discontinuities make the representation have significant higher harmonic content. This is a very inelegant representation. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

CTFS of Even and Odd Functions M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Convergence of the CTFS Partial CTFS Sums For continuous signals, convergence is exact at every point. A Continuous Signal M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Convergence of the CTFS Partial CTFS Sums For discontinuous signals, convergence is exact at every point of continuity. Discontinuous Signal M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Convergence of the CTFS At points of discontinuity the Fourier series representation converges to the mid-point of the discontinuity. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Numerical Computation of the CTFS How could we find the CTFS of a signal which has no known functional description? Numerically. Unknown M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Numerical Computation of the CTFS Samples from x(t) M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Numerical Computation of the CTFS M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Numerical Computation of the CTFS M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Properties Linearity M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Properties Time Shifting M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 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 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Properties Time Scaling (continued) M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Properties Change of Representation Time M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Properties Time Differentiation M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Properties Time Integration Case 1 Case 2 is not periodic M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Some Common CTFS Pairs M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Examples M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Examples M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFS Examples M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

LTI Systems with Periodic Excitation M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

LTI Systems with Periodic Excitation M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

LTI Systems with Periodic Excitation M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

LTI Systems with Periodic Excitation M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 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 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

CTFS-to-CTFT Transition M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

CTFS-to-CTFT Transition M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

CTFS-to-CTFT Transition M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

CTFS-to-CTFT Transition M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

CTFS-to-CTFT Transition M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Definition of the CTFT M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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 in time. (Time limited means “having non-zero values only for a finite time.”) M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Frequency Content Highpass Lowpass Bandpass M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Some CTFT Pairs M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Convergence and the Generalized Fourier Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Convergence and the Generalized Fourier Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Convergence and the Generalized Fourier Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Convergence and the Generalized Fourier Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Convergence and the Generalized Fourier Transform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Negative Frequency This signal is obviously a sinusoid. How is it described mathematically? M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Negative Frequency M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl Negative Frequency M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl More CTFT Pairs M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFT Properties Linearity M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFT Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFT Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFT Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The “Uncertainty” Principle The 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. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFT Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFT Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFT Properties In the frequency domain, the cascade connection multiplies the frequency responses instead of convolving the impulse responses. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFT Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFT Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFT Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFT Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFT Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl CTFT Properties M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Numerical Computation of the CTFT M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl