frequency response and continuous-time fourier...
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Frequency Response andContinuous-time Fourier Transform
Signals and Systems in the FD-part II
Goals
I. (Finite-energy) signals in the Frequency Domain- The Fourier Transform of a signal- Classification of signals according to their spectrum(low-pass, high-pass, band-pass signals)- Fourier Transform properties
II. LTI systems in the Frequency Domain- Impulse Response and Frequency Response relation- Computation of general system responses in the FD
III. Applications to audio signals- A simple design of an equalizer
Objectives
How can we extend the Fourier Series method to other signals?
There are two main approaches:
The Fourier Transform (used in signal processing)
The Laplace Transform (used in linear control systems)
The Fourier Transform is a particular case of the LaplaceTransform, so the properties of Laplace transforms areinherited by Fourier transforms. One can compute Fouriertransforms in the same way as Laplace transforms.
Fourier Transform
Generalization of Fourier Series to aperiodic functions
Recall, for a periodic function of period ,
For an aperiodic function, take
Requires x(t) to be absolutely integrable
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x(t) = X[k]ejk" 0t
k=#$
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T0
x(t)e" jk# 0tdt
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Inverse Fourier transform Fourier transform
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Relationship with Laplace transform
The Fourier Transform is a particular case of Laplace transform
is often called spectrum or amplitude spectral density(spectral refers to ‘ variation with respect to frequency’,density refers to ‘amplitude per unit frequency’)
Sometimes, the transform is seen as a function of cyclicfrequency . That is,
!
F( j") = g(t)e# j"t
dt#$
$
%
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F( j") = L( j")
!
L(s) = g(t)e"stdt
"#
#
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2"f =#
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F( j") = F( j2#f ) = F( f )
!
F( j") = F( j" /2# ) = F( f )
!
F( j" )
Transform methods get more powerful
Fourier series to Fourier transform to Laplace transform
A finite-amplitude, real signal can be represented as
- periodic case: countable infinity of complex exponentials
- aperiodic case: uncountable infinity of complex exponentials
Important features may be obscure in TD, but crystalline in FD.
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k=#$
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x(t) =1
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d#$%
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General signals in the Frequency Domain
The signal representation in the Frequency Domain is thegraph of and of
One can classify signals into:
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|F( j") |
Low-pass signal: The energy of the signal is concentratedat low frequencies (these signals have “slow transitions”in the time domain and are smooth)
High-pass signal: The energy of the signal is concentratedat high frequencies (these signals have zero mean andrapidly changing values; e.g. noise)
Band-pass signal: The spectrum vanishes at low and highfrequencies and concentrates in an intermediatefrequency band (signals that look more like a puresinusoid with high frequency)
!
"F( j#) = arctan(Im(F( j#) /Re(F( j#))
Graphical Interpretation TD versus FD
Examples using cyclic frequency:
Low-pass High-pass
Band-pass
Examples: computation of FT
Let’s compute the FT of
Let’s compute the FT of
!
x(t)= e"a | t |
, Re(a) > 0
!
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dt#$
$
% = e#a | t |e# j"t
dt#$
$
% = eat# j"t
dt#$
0
% + e#at# j"t
dt0
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%
=1
a # j"e
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a + j"e#(a+ j" )t
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=1
a # j"+
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x(t)=1
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X( j" ) = x(t)e# j"t
dt#$
$
% = e# j"t
dt#$
$
% = ? Integral does not converge
Computation of FT
What do we do? Calculate FT of and then take
Now,
Therefore,
Similarly, one can also compute
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x(t)e"#t
, # > 0
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" #0
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" 2+# 2
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lim"# 0
X" ( j$ ) = %, if $ = 0
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X" ( j# )d# = 2$%&
&
'
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X( j" ) = 2#$(")
!
cos(at)"#($(% & a) +$(% + a))
!
sin(at)" j# ($(% + a) &$(% & a))
Graphical representation
Observe this is the same as the Fourier Series spectrum graphs!
TD and FD representation of a signal
The Time Domain and Frequency Domain representations areused to emphasize different aspects
TD graph: you can see when something occurs and theamplitude of occurrence. For example, this is good fordescribing step or discontinuous signals
FD graph: the magnitude and phase gives you informationabout what produces the oscillations present in the signal.Good for describing audio signals
Example: sound from ringing a wine glass.The harmonics, fundamental frequency, tellsus the nature of the material that producedit. This information is not contained in asingle sample time of a TD description
TD versus FD: what to choose?
In general, the use of TD or FD signal representations dependson the application you are working with
For ECG monitors, you want to emphasize time occurrences,and produce filters that generate good step responses*. Soyou study time domain representations of step responses
For a hearing aid, you want to emphasize certain frequencyranges, so you want to optimize your filter with respect tofrequency responses. So you study frequency domainrepresentations of audio signals
Filters that perform well in terms of frequencies do not performwell in terms of step responses, and the other way around
*a good step response should have a shape as close to a step as possible
Fourier Transform pairs Table
A few more examples of Fourier Transform pairs from the book(Table 10.1, observe there are two columns, one for andone for )
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f
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sinc(t)" rect(# /2$ )
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Fourier Transform properties (radian freq)
LinearityShift in timeTime scalingFrequency scalingFrequency shiftingModulation
Differentiation
Integration
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ax(t) + by(t)" aX( j#) + bY ( j#)
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x(t)cos"0t#1
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1
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a)" X(aj#)
Example: calculate FT of a periodic funtion
Let be periodic of period
Using linearity and frequency s hifting,
In other words,the Fourier transform is a generalization ofthe Fourier series
!
x(t)
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T0
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x(t) = X[k]ejk" 0t
k=#$
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%
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X( j" ) = 2# X[k]k=$%
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& '(" $ k"0)
Convolution property of Fourier transforms
Convolution/Multiplication duality:
In particular we have!
h(t)" f (t) = h(t #$) f ($)d$#%
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& 'H( j()F( j()
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h(t)F
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h(t) f (t)"1
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What are the implications?
For a stable linear system we can compute system responses as
, are signal Fourier transforms is the system frequency response
Spectra
The Frequency Response contains all the information needed todetermine any system response
Deconvolutions can be computed by division in the FD:If , then
(can be done for invertible systems)
!
Y j"( ) = H j"( )X j"( )
!
Y ( j") = H( j") X( j"), #Y ( j") =#H( j") +#X( j")
!
y(t) = h(t) * x(t)
!
x(t) = F"1{Y ( j#) /H( j#)}
!
Y j"( )
!
X j"( )
!
H j"( )
Signals/systems in the FD
Similarly to the Fourier Series case, once the input signal andsystem Fourier transforms are computed, the responsecan be obtained through simple multiplication and sumoperations
Additionally, the FT/IFT can be approximated in a computerthrough special routines: the FFT (Fast Fourier Transform)and the IFFT (Inverse Fast Fourier Transform)
Overall, this makes the whole process in the FD muchfaster than convolution in the TD. The only disadvantageis that the FT only applies to finite energy signals
Signals/systems in the FD
GoalsI. (Finite-energy) signals in the Frequency Domain
- The Fourier Transform of a signal- Classification of signals according to their spectrum (low-
pass signals, high-pass signals, band-pass signals)- Fourier Transform properties
II. LTI systems in the Frequency Domain- Impulse Response and Frequency Response relation- Computation of general system responses in the FD
III. Applications to audio signals- A simple design of an equalizer
A simple design of an equalizer
Applications of the Fourier Transform theory are many. One ofthem is the design of equalizers to accomplish differenteffects on audio signals:
-Bass volume or low frequencies in your audio system canbe implemented using a low-pass filter;
-Treble volume or high frequencies can be implementedusing a high-pass filter
-The previous two filters can be combined together tofilter only those sounds with frequencies around a value of
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H1(") =
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j" +"c
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H2(") =
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j" +"c
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j" +"c
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j" +"c
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"c
A simple design of an equalizer
Example graphs of these filters for a given values ofthe cutoff frequencies :
!
"c# lowpass = 50000,
!
"c#highpass =100
A simple design of an equalizer
Another example of band-pass filter centered at agiven frequency and whose magnitude canbe adjusted is the following:
(Each color corresponds to a different choice of )
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( j")2 + j2"0" /10# +"
0
2
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0
2
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=1
!
"
A simple design of an equalizer
A simple equalizer can be built by “connecting in series” band-pass filters like the previous one as follows:
The center frequencies of each band-pass filter are 20Hz, 30Hz,40Hz,…. The Frequency Response of the equalizer is obtainedby summing all the band-pass filters’ FR. By choosingdifferent values of the parameters beta, one can emphasizeor de-emphasize any frequency range in an audio signal
Summary
Important points to remember:
A signal can be classified into a low-pass, high-pass or band-pass signaldepending on its magnitude and phase spectrum
In the time domain, low-pass signals correspond to signals with slowtransitions. High-pass signals correspond to signals with fasttransitions. Band-pass signals look like sinusoids/co-sinusoids.
The Fourier Transform theory allows us to extend the techniques andadvantages of Fourier Series to more general signals and systems
In particular we can compute the response of a system to a signal bymultiplying the system Frequency Response and the signal FourierTransform. (And we can avoid convolution)
The Fourier Transform of the Impulse Response of a system is preciselythe Frequency Response
The Fourier Transform theory can be used to accomplish different audioeffects, e.g. the design of equalizers