The Continuous-Time Fourier Series
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Representing a Signal
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Linearity and Superposition If an excitation can be expressed as a linear combination of complex sinusoids, the response of an LTI system can be expressed as a linear combination of responses to complex sinusoids.
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Real and 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 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.)
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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.
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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. The frequency of this second sinusoid is the second harmonic of the fundamental frequency.
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Conceptual Overview The best approximation to the dashed-line signal using a constant plus three harmonics is the solid line. In this case (but not in general), the third harmonic has zero amplitude. This means that no sinusoid of three times the fundamental frequency improves the approximation.
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Conceptual Overview The best approximation to the dashed-line signal using a constant plus four harmonics is the solid line. This is a good approximation that gets better with the addition of more harmonics.
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Continuous-Time Fourier Series Definition
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Orthogonality
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Orthogonality
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 all the component functions are based on the same representation 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 #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. Notice that the magnitude is an even function of k and the phase is an odd function of k.
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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 #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. Although correct in the time interval from zero to 7.5 ms, 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
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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 discontinuity the Fourier series representation converges to the mid-point of the discontinuity.
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Numerical Computation of the CTFS How could we find the CTFS of a signal that has no known functional description?
Numerically.
Unknown
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Numerical Computation of the CTFS
Samples from x(t)
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Numerical Computation of the CTFS
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Numerical Computation of the CTFS
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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 Properties Time Scaling (continued)
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CTFS Properties
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CTFS Properties Change of Representation Time
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CTFS Properties
Time Differentiation
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Time Integration
is not periodic
CTFS Properties
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
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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LTI Systems with Periodic Excitation
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LTI Systems with Periodic Excitation
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LTI Systems with Periodic Excitation
The Continuous-Time Fourier Transform
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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 (and also a periodic) signal for all time
CTFS-to-CTFT Transition
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CTFS-to-CTFT Transition
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CTFS-to-CTFT Transition
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CTFS-to-CTFT Transition
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CTFS-to-CTFT Transition
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Definition of the CTFT
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Definition of the CTFT
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Definition of the CTFT
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Some Remarkable Implications of the Fourier Transform
The CTFT can express 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-valued sinusoids, each of which is unlimited in time. (Time limited means “having non-zero values only for a finite time.”)
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Frequency Content
Some CTFT Pairs
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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
More CTFT Pairs
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Negative Frequency This signal is obviously a sinusoid. How is it described mathematically?
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Negative Frequency
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Negative Frequency
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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” 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.
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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 multiplies the frequency responses instead of convolving the impulse responses.
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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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