Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a...

30
DR. TAREK TUTUNJI PHILADELPHIA UNIVERSITY, JORDAN 2014 Fourier Series and Fourier Transforms

Transcript of Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a...

Page 1: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

D R . T A R E K T U T U N J I

P H I L A D E L P H I A U N I V E R S I T Y , J O R D A N

2 0 1 4

Fourier Series and Fourier Transforms

Page 2: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Fourier Series

The frequency representation of periodic and aperiodic signals indicates how their power or energy is allocated to different frequencies. Such a distribution over frequency is called the spectrum of the signal. For a periodic signal the spectrum is discrete, as its power is concentrated at

frequencies multiples of a so-called fundamental frequency, directly related to the period of the signal. On the other hand, the spectrum of an aperiodic signal is a continuous function of frequency.

The Fourier representation is also useful in finding the frequency response of linear time-invariant systems, which is related to the transfer function obtained with the Laplace transform.

The frequency response of a system indicates how an LTI system responds to sinusoids of different frequencies. Such a response characterizes the system and permits easy computation of its steady-state response, and will be equally important in the synthesis of systems.

Fourier analysis is in the steady state, while Laplace analysis considers both transient and steady state

Page 3: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Frequency Response

Page 4: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

RC Example

Consider the RC circuit.

Let the voltage source be vs(t)=4 cos(t+pi/4) volts, the resistor be R = 1Ohm, and the capacitor C =1 F.

Find the steady-state voltage across the capacitor.

Page 5: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

RC Example

Page 6: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Fourier Series

Page 7: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Fourier Series

The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency multiples of the fundamental frequency of x(t) The advantage of using the Fourier series to represent

periodic signals is not only the spectral characterization obtained, but in finding the response for these signals when applied to LTI systems.

The Fourier series can be used to represent signals by a combination of different sines and cosines

Page 8: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Fourier Series: Trigonometric

Page 9: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Example

Page 10: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Example

Page 11: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Power Spectrum

Page 12: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Symmetry

Page 13: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Example

Page 14: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Example

Page 15: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Basic Properties of Fourier Series

Page 16: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Fourier Transform

Page 17: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Fourier Transform and Laplace

Page 18: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Notes

If x(t) has a finite time support and in that support x(t) is finite, its Fourier transform exists. To find it use the integral definition or the Laplace transform of x(t)

If x(t) has a Laplace transform X(s) with a region of convergence including the jW, its Fourier transform is X(s) at s = jW•.

If x(t) is periodic of infinite energy but finite power, its Fourier transform is obtained from its Fourier series using delta functions.

Consider the Laplace transform if the interest is in transients and

steady state, and the Fourier transform if steady-state behavior is of interest.

Represent periodic signals by their Fourier series before considering their Fourier transforms.

Attempt other methods before performing integration to find the Fourier transform.

Page 19: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Example

Page 20: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Example

Page 21: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Example

Page 22: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Fourier Transform of Periodic Signals

Page 23: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Example

Page 24: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Example

Page 25: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Fourier Transform Example

Page 26: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Fourier Transform Example

Page 27: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Basic Properties

Page 28: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Convolution and Filtering

Page 29: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Conclusion

The Fourier series can be used to represent a signal by a combination of different sin and cos signals

Fourier Transforms are used to provide information about the frequency behavior of signals.

The frequency analysis provide the fundamentals for filter design.

There is a well-established relationship between the Fourier and Laplace Transforms.

Page 30: Fourier Series and Fourier Transforms · Fourier Series The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency

Reference

Chapter 4 and 5, Signals and Systems using MATLAB by Luis Chaparro. Elsevier Publisher 2011