Document 6520193

What is Fourier
Series?
A Fourier series is an expansion of a periodic function in terms of an infinite sum of
sines and cosines. Fourier series make use of the orthogonality relationships of the
sine and cosine functions. The computation and study of Fourier series is known as
harmonic analysis and is extremely useful as a way to break up an arbitrary periodic
function into a set of simple terms that can be plugged in, solved individually, and
then recombined to obtain the solution to the original problem or an approximation
to it to whatever accuracy is desired or practical. Examples of successive
approximations to common functions using Fourier series are illustrated above.
Why do We Need Fourier
Analysis?
Harmonically Related Complex Exponentials
(cont’d)
• The essence of Fourier analysis is to represent signals in
terms of complex exponentials (or trigonometric
functions)
x(t ) =
∞
∑ ak e
jω 0 t
k = −∞
= ... + a− 2e
jω0t
+ a−1e
jω 0 t
+ a0 + a1e
jω0t
+ a2 e
jω0t
•
x(t ) =
+ ...
• Many reasons:
– Almost any signal can be represented as a series of complex
exponentials
– A compact way of approximating several signals. This opens a
lot of applications:
• storing analog signals (such as music) in digital environment
• over a digital network, transmitting digital equivalent of the signal
instead of the original analog signal is easier!
Thus, a linear combination of them is also periodic!:
∞
∑C e
k = −∞
•
•
•
•
k
jkω0t
So, that means it should be possible to split a periodic signal into a
set of periodic signals with same fundamental frequency.
This above representation of a periodic signal is referred as Fourier
Series representation of that signal.
In the Fourier series above, the terms for k=1 and k=-1 are referred
as the fundamental components or the first harmonic components of
x(t).
There are a number of different forms a Fourier Series can take. They
are equivalent. Engineers work with the Exponential form.
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Fourier Series Coefficients
Find its Fourier Series coefficients
Ck =
Example 1 Given a signal y(t) = cos(2t),
find its Fourier Series coefficients.
Example : Calculate the Fourier Series coefficients for the impulse train.
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x(t )e − jkω0t dt
T T∫
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Example : Calculate the Fourier Series of a square wave .
Chapter-5 Fourier Transform
In the previous three lessons, we discussed the
Fourier Series, which is for periodic signals.
This lesson will cover the Fourier Transform
which can be used to analyze aperiodic signals.
(Later on, we'll see how we can also use it for
periodic signals.)
The Fourier Transform is another method for
representing signals and systems in the
frequency domain
the ω-axis, distance between two consecutive aks is now ω0=2π/T, the
fundamental frequency.
„On
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Bridge Between Fourier Series and
Transform
• Consider the periodic signal x(t) below:
x(t)
-T
t
-T1 0 T1
T
• We know that the Fourier coefficients for x(t) will be:
 2T1
,
 T
ak = 
 sin(kω 0T1 )

kπ
Bridge Between Fourier Series and Transform
(cont’d)
• Now, sketch ak on the ω-axis:
ak
2T1/T
-2ω0 -ω0
0 ω0 2ω0
ω
• On the ω-axis, distance between two consecutive aks is
now ω0=2π/T, the fundamental frequency.
k =0
k ≠0
Bridge Between Fourier Series and Transform
(cont’d)
• As the period TÆ∝, the fundamental frequency ω0Æ0.
So, the distance between the two consecutive aks
becomes zero, and the sketch of ak becomes continuous,
what is called as Fourier Transform.
• At the other side, as TÆ∝, the signal x(t) becomes
aperiodic and takes the form:
x(t)
-T1 0 T1
t
• This means the Fourier Transform can represent an
aperiodic signal on the frequency-domain.
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Example Find the Fourier Transform of cos(ω0t)
Multiplication of Signals
• It states that the Fourier Transform of the product of
two signals in time is the convolution of the two
Fourier Transforms.
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Proof
Therefore,
Fourier Transforms of Sampled Signals
In this lesson, we will discuss sampling of
continuous time signals. Sampling a
continuous time signal is used, for
example, in A/D conversion, such as
would be done in digitizing music for
storage on a CD, digitizing a movie for
storage on a DVD, or taking a digital
picture.
Now, we will derive the Sampling Theorem . To do this, we will
examine our signals in the frequency domain.
To start, let p(t) have a Fourier Transform P(ω), x(t) have a
Fourier Transform X(ω), and xs(t) have a Fourier Transform
Xs(ω).
Then, because xs(t) = x(t)p(t), by the Multiplication
Property,
Now let's find the Fourier Transform of p(t). Because
the infinite impulse train is periodic, we will use the
Fourier Transform of periodic signals:
where Ck are the Fourier Series coefficients of the periodic signal.
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Let's find the Fourier Series coefficients Ck for the periodic
impulse train p(t):
Therefore
Now, to finish our derivation of the Sampling Theorem,
we will go back and determine Xs(ω).
We saw that:
Therefore,
or we get replicated, scaled versions of X(ω), spaced every
ω0 apart in frequency:
From this development and observing the above figure, we can
derive our Sampling Theorem. This is one of the most important
results.
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We can recover x(t) from its sampled version xs (t) by
using a low pass filter to recover the center island:
After low pass filtering
As you can see from the figure if ω0 - ωc < ωc, we would get overlap of the
replicates of X(ω) in frequency. This is known as "aliasing." Therefore, to
avoid aliasing, we require ω0 - ωc > ωc or ω0 > 2ωc. If we avoid aliasing,
we can recover x(t) from its samples. (Usually, we choose a sampling rate a
bit higher than twice the highest frequency since filters are not ideal.)
We hear music up to 20 kHz and CD sampling rate is 44.1 kHz. (Dogs
would need a higher quality CD since they hear higher frequencies than
humans.)
Sampling Theorem Conditions
Sampling Theorem: A continuous-time signal x(t) can be
uniquely reconstructed from its samples xs(t) with two
conditions:
– x(t) must be band-limited with a maximum frequency ωM
– Sampling frequency ωs of xs(t) must be greater than 2ωM, i.e.
ωs>2ωM.
• The second condition is also known as Nyquist Criterion.
• ωs is referred as Nyquist Frequency, i.e. the smallest
possible sampling frequency in order to recover the
original analog signal from its samples.
Aliasing (Under-sampling)
• What happens when sampling frequency is less than
Nyquist Frequency, i.e. ωs<2ωM ?
• The original signal x(t) cannot be recovered from xs(t)
since there will be unwanted overlaps in Xs(ω).
• This will cause the recovered signal xr(t) to be different
than x(t), also called as aliasing or under-sampling.
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Example 2 The inverse Fourier Transform of the signal in the
previous example is
Draw the sampled signals using the sampling trains of the
previous example
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Example 2 AM radio - Double-sideband, suppressed
carrier, amplitude modulation
•
Let x(t) be a music signal with
Fourier Transform X(ω).
To modulate this signal, we'll form and transmit
Your radio demodulates the signal by multiplying the
received signal y(t) by another cosine wave to form a third
signal z(t):
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To recover the original music signal x(t) from the demodulated
signal z(t), you filter z(t) with a low-pass filter by multiplying by a
rect function in frequency. This corresponds to convolving with a
sinc function in time.
The Bilateral Laplace Transform of a signal x(t) is defined as:
The complex variable s = σ + jω, where ω is the frequency
variable of the Fourier Transform (simply set σ = 0). The
Laplace Transform converges for more functions than the
Fourier Transform since it could converge off of the jω axis.
Here is a plot of the s-plane:
The xy-axis plane, where x-axis
is the real axis and y-axis is the
imaginary axis, is called as splane. We will perform our
visualizations of Laplace
transform on s-plane
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Region of Convergence (ROC)
• Similar to the integral in Fourier transform, the integral in
Laplace transform may also not converge for some
values of s.
• So, Laplace transform of a function is always defined by
two entities:
– Algebraic expression of X(s).
– Range of s values where X(s) is valid, i.e. region of convergence
(ROC).
bilateral and unilateral Laplace
Transforms
As we'll see, an important difference between the
bilateral and unilateral Laplace Transforms is that you
need to specify the region of convergence (ROC) for
the bilateral case.
We point out (without proof) several features of ROCs:
• A right-sided time function (i.e. x(t) = 0, t < t0 where
t0 is a constant) has an ROC that is a right half-plane.
• A left-sided time function has an ROC that is a left
half-plane.
• A 2-sided time function has an ROC that is either a
strip or else the ROC does not exist, which means that
the Laplace Transform does not exist.
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or
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Example
Example
Recall the equation for the voltage of an inductor:
If we take the Laplace Transform of both sides of this equation, we get:
which is consistent with the fact that an inductor has impedance sL
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Example 1 Find y(t) where the transfer function H(s) and the input x(t)
are given. Use Partial Fraction Expansion to find the output y(t):
Example 5 Find the output of an LTI system with impulse
response h(t) = ebtu(t) to an input x(t) = eatu(t), where a ≠ b.
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