Lecture 4 Spectrum Representation

Fundamentals of Digital Signal Processing

Spring, 2012

Wei-Ta Chu
2012/3/6

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 Spectrum (頻譜)
Spectrum: a compact representation of the frequency
content of a signal that is composed of sinusoids.
We will show how more complicated waveforms can
be constructed out of sums of sinusoidal signals of
different amplitudes, phases, and frequencies.
A signal is created by adding a constant and N
sinusoids

Such a signal may be represented in terms of the

complex amplitude

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 Spectrum

By using inverse Euler formula

The real part of a complex number is equal to one-half

the sum of that number and its complex conjugate.
Each sinusoid in the sum decomposes into two rotating
phasors, one with positive frequency, , and the
other with negative frequency, .

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 Spectrum
Two-sided spectrum of a signal composed of sinusoids
as

Set of pairs

Each pair indicates the size and relative phase of the

sinusoidal component contributing at frequency
Frequency-domain representation of the signal

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 Example

Apply inverse Euler formula

The constant component of the signal, often called the

DC component, can be expressed as .
The spectrum of the signal is represented by

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 Notation Change
Introduce a new symbol for the complex amplitude in
the spectrum

The spectrum is the set of (fk, ak) pairs

Define f0 = 0. Now we can rewrite

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 Graphical Plot of the Spectrum
Spectrum plot of

This plot shows the relative location of the frequencies, and

the relative amplitudes of the sinusoidal components
The rotating phasors with positive and negative frequency
must combine to form a real signal

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 Spectrum
A general procedure for computing and plotting the
spectrum for an arbitrarily chosen signal requires the
study of Fourier analysis.
If it’s known a priori that a signal is composed of a
finite number of sinusoidal components, the process of
analyzing that signal to find its spectral components
involves writing an equation for the signal in the form
of

and picking off the amplitude, phase, and frequency of

each of its rotating phasor components

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 Beat Notes
When we multiply two sinusoids having different
frequencies, we can create an audio effect called beat
note.
The phenomenon is best heard by picking one of the
frequencies to be very small (e.g., 10 Hz), and the
other around 1kHz.
One use for multiplying sinusoids is modulation for
radio broadcasting. AM radio stations use this method,
which is called amplitude modulation. (調幅)

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 Multiplication of Sinusoids
Example: Spectrum of a product

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 Multiplication of Sinusoids

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 Multiplication of Sinusoids

Four terms are in the additive combination. The four

spectrum components are at 5.5, 4.5, -4.5, and -5.5 Hz
Note that neither of the original frequencies (5Hz and
½ Hz) used to define x(t) are in the spectrum.
Product of two sinusoids is equivalent to a sum.
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 Beat Note Waveform
We want to derive a general relationship between any
beat signal, its spectrum, and the product form if we
start with an additive combination of two closely
spaced sinusoids.

The two frequencies can be expressed as

and , where we have defined a center
frequency and a derivation
frequency , which we assume is much
smaller than fc

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 Beat Note Waveform
Rewrite x(t) as a product of two cosines, and thereby
have a form that is easier to plot in the time domain.

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 Example

envelope

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 Example
The effect of multiplying the higher-frequency
sinusoid (200 Hz) by the lower-frequency sinusoid (20
Hz) is to change the envelope of the peaks of the
higher-frequency waveform.

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 Example
If is decreased to 9 Hz, the envelope of the 200 Hz
tone changes much more slowly.
The time interval between nulls of the envelope is
The more closely spaced the frequencies in

the slower the envelope

variation
smaller

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 Example

The spectrum for x(t) contains frequency components

at Hz and Hz
Musician use this beating
phenomenon as an aid in tuning
two instruments to the same pitch.

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 Amplitude Modulation
Amplitude modulation is the process of multiplying a
low-frequency signal by a high-frequency sinusoid. It’s
the technique used to broadcast AM radio.
The AM signal is a product of the form

It’s assumed that the frequency of the cosine term (fc

Hz) is much higher than any frequencies contained in
the spectrum of v(t), which represents the voice or
music signal to be transmitted.
The cosine wave is called the carrier
signal (載波訊號), and its frequency is called the
carrier frequency.

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 Example

Multiplying the higher-frequency sinusoid by the

lower-frequency sinusoid is to “modulate” (or change)
the amplitude envelope of the carrier waveform.

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 Amplitude Modulation
The primary difference between this AM signal and the
beat signal is that the envelope never goes to zero.
The DC component (5) is greater than the amplitude (4)
of the 20-Hz component.
When the carrier frequency becomes very high
compared to v(t), it’s possible to see the outline of the
modulating cosine without drawing the envelope
signal explicitly.

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 Amplitude Modulation
The AM signal spectrum is nearly the same as the beat
signal. The only difference being a relatively large
term at f=fc.

There are six spectral components of x(t) at the

frequencies Hz and Hz, and also at the
carrier frequency Hz.

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 Exercise 3.2

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 Periodic Waveforms
A periodic signal satisfies the condition that
for all t, which states that the signal repeats its values
every T0 secs.
T0 is called the period. If it is the smallest such
repetition interval, it is called fundamental period. (基
本週期)
E.g. has a period of ½ sec, but its
fundamental period is T0=1/4 sec.

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 Periodic Waveforms
Periodic signals can be synthesized by adding two or
more cosine waves that have harmonically (調和的)
related frequencies, i.e. all frequencies are integer
multiples of a frequency f0.

(harmonic frequencies)

The frequency fk is called the kth harmonic of f0

because it’s an integer multiple of the basic frequency
f0, which is called the fundamental frequency. (基本頻
率, 基頻)

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 Fundamental Frequency
The fundamental frequency is the largest f0 such that
fk=kf0.
This is the greatest common divisor:

E.g. if the signal is the sum of sinusoids with

frequencies 1.2, 2, and 6 Hz, then f0 = 0.4 Hz. 1.2 Hz is
the 3rd harmonic (泛音), 2 Hz is the 5th harmonic, and
6 Hz is the 15th harmonic.

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 Periodic Waveforms

Since each cosine has a period of 1/f0, the sum must

have exactly the same period and x(t+1/f0) = x(t).
The period of x(t) is T0 = 1/f0. Since T0=1/f0 is the
smallest possible period, it’s also the fundamental
period of x(t).
Using the complex exponential representation

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 Synthetic Vowel
An example of synthesizing a periodic signal, consider
a case where the sum contains nonzero terms for
, and where f0 = 100 Hz.
This signal approximates the waveform produced by a
man speaking the vowel sound “ah”.

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 Synthetic Vowel
All the frequencies are integer multiples of 100 Hz,
even though there is no spectral component at 100 Hz
itself.
Compare magnitude spectrum with phase angle
spectrum
Complex amplitudes of the
negative frequencies are the
complex conjugates of the
complex amplitude for the
corresponding positive
frequencies.

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 Synthetic Vowel
Five sinusoidal terms
Successively plot the waveforms
corresponding to these sinusoids
Both x2(t) and x4(t) have the period
5 msec. (fund. frequency = 200 Hz)
In x5(t), the period increases to 10
msec, because 200, 400, and 500
Hz are integer multiples of 100 Hz.
The high frequencies contribute the
fine detail in the waveform –
increasingly complicated after
adding the 16th and 17th harmonics.

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 Demo

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