Transcript Qassim University College of Engineering Electrical

Qassim University
College of Engineering
Electrical Engineering Department
Course: EE301: Signals and Systems Analysis
Part C
Instructor: Associate Prof. Dr. Ahmed Abdelwahab
Fourier Spectrum
The compact trigonometric Fourier series indicates that a
periodic signal g(t) can be expressed as a sum of sinusoids
of frequencies 0 (dc), ωo, 2ωo, ... , nωo, . ., whose
amplitudes are Co, CI, C2, . . ., Cn,.. . and whose phases
are 0, θ1, θ2, . . ., θn, ....
• Amplitude spectrum is a plot of the amplitude Cn,vs. ω.
• Phase spectrum is a plot of the phase θn, vs. ω.
These two plots together are the frequency spectra of g(t).
Note that the frequency spectra of a periodic signal is a
discrete-frequency signal.
These spectra tell us at a glance the frequency
composition of g(t), that is, the amplitudes and
phases of various sinusoidal components of
g(t). Knowing the frequency spectra, g(t) can
be reconstructed or synthesized.
Therefore, the frequency spectra provide an
alternative description that is the frequency-domain
description of g(t) beside The time-domain
description. It can be shown that the Fourier series of
any even periodic function g(t) consists of cosine
terms only and the series of any odd periodic function
g(t) consists of sine terms only.
EXPONENTIAL FOURIER SERIES
The form of the exponential series is more
compact. Moreover, the mathematical
expression for deriving the coefficients of the
series is also compact. It is much more
convenient to handle the exponential series
than the trigonometric one. In the system
analysis also, the exponential form proves
more convenient than the trigonometric form.
Exponential Fourier Spectra
It follows that the amplitude spectrum (IDnI vs. ω) is an even function of ω and
the angle spectrum (θn vs. ω) is an odd function of ω when g(t) is a real signal.
What Is a Negative Frequency?
The existence of the spectrum at negative frequencies is
somewhat disturbing because by definition, the frequency
(number of repetitions per second) is a positive quantity.
How do we interpret a negative frequency?
Using a trigonometric identity, the sinusoid of a negative
frequency –ω0 can be expressed as
APERIODIC SIGNAL REPRESENTATION
BY FOURIER INTEGRAL
Electrical engineers instinctively think of signals in terms of their
frequency spectra and think of systems in terms of their frequency
responses. This is basically thinking in the frequency domain.
spectral representation of periodic signals (Fourier series) is
discussed. Now, spectral representation can be extended to aperiodic
signals. To represent an aperiodic signal g(t), such as the one shown
in next slide Fig. (a) by everlasting exponential signals, let us
construct a new periodic signal gTo (t) formed by repeating the signal
g(t) every To seconds, as shown in the same Fig. (b). The period To
is made long enough to avoid overlap between the repeating pulses.
The periodic signal gTo(t) can be represented by an exponential
Fourier series. If we let To → ∞ , the pulses in the periodic signal
repeat after an infinite interval, and therefore
TRANSFORMS OF SOME USEFUL FUNCTIONS
Symmetry of Direct and Inverse Transform
Operations-Time-Frequency Duality
An interesting fact: the direct and the inverse transform operations
are remarkably similar. It is the basis of the so-called duality of time
and frequency. The duality principle may be compared with a
photograph and its negative.
Observe, for example, the role reversal of time and frequency in
these two equations (with the minor difference of the sign change in
the exponential index).
The value of this principle lies in the fact that
whenever we derive any result, we can be sure that it has a dual
Symmetry Property
Note that rect (- x)
= rect (x) because
rect is an even
function.
Scaling Property
The function g(at) represents the function g(t) compressed in time by a factor a. Similarly, a
function G(ω/a) represents the function G(ω) expanded in frequency by the same factor a. The
scaling property states that time compression of a signal results in its spectral expansion, and
time expansion of the signal results in its spectral compression. The scaling property implies
that if g(t) is wider, its spectrum is narrower, and vice versa.
Doubling the signal duration halves its bandwidth, and vice versa. This suggests that the
bandwidth of a signal is inversely proportional to the signal duration or width (in seconds).
Time-Shifting Property
This result shows that delaying a signal by to seconds does not change its
amplitude spectrum. The phase spectrum, however; is changed by -ωto.
This would give the following interesting transform pair:
Physical Explanation of the Linear Phase
Time delay in a signal causes a linear
phase shift in its spectrum. Imagine g(t)
being synthesized by its Fourier
components, which are sinusoids of
certain amplitudes and phases. The
delayed signal g(t - to) can be synthesized
by the same sinusoidal components, each
delayed by to seconds. The amplitudes of
the components remain unchanged.
Therefore, the amplitude spectrum of g(t - to) is identical to that of g(t).
The time delay of to in each sinusoid, however, does change the phase of each component.
A time delay to in a sinusoid of frequency ω manifests as a phase delay of ωto. This is a
linear function of ω, meaning that higher frequency components must undergo
proportionately higher phase shifts to achieve the same time delay.
This effect is shown in the above Fig. with two sinusoids, the frequency of the lower
sinusoid being twice that of the upper. The same time delay td amounts to a phase shift of
π/2 in the upper sinusoid and a phase shift of π in the lower sinusoid. This verifies the fact
that to achieve the same time delay, higher frequency sinusoids must undergo
proportionately higher phase shifts.
Frequency-Shifting Property
Shifting the Phase Spectrum of a Modulated Signal
The Fourier transform of a general
periodic signal g(t) of period To
the impulse train
can be expressed as an
exponential Fourier series as
Here Dn = 1/ TO. Therefore,
Thus, the spectrum of the impulse train also
happens to be an impulse train (in the frequency
domain), as shown above.
Convolution
Time Differentiation and Time Integration
Time Integration