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Continuous-Time Fourier Methods

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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

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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.

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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.

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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.

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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.

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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 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.

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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. 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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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 which 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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CTFS Properties

Time 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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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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Continuous-Time Fourier Methods

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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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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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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.”)

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Frequency Content

Lowpass

Highpass

Bandpass

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Some CTFT Pairs

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