9.4 discrete fourier transform (dft)eceweb1.rutgers.edu/~gajic/solmanual/slides/chapter9_dft.pdf ·...
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9.4 Discrete Fourier Transform (DFT)
The discrete-timeFourier transform,DTFT, can be derivedalso in the processof
numericalevaluationof the integral that definesthe Fourier transform. Consider
the basicdefinition of the Fourier transform�
� ��������
andapproximateit by an infinite sumobtainedby performingsampling(discretiza-
tion of the time axis) with the samplingperiod � . In sucha case,we have
��
� � � � ����� �� �� � �
� ��
� � ���� ��� �
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or
������� � � ������� �
In order to be able calculatethe DTFT sum numerically (using a computer),we
haveto approximatetheinfinite time durationsignal by its finite time duration
approximation.In that direction,we first define the DTFT of length as�! #"$&% �
�' $�(
This approximationis meaningfulif the signalvaluesareconcentratedin the time
interval andthoseoutsideof this interval havenegligiblevalues.Since
can be arbitrary chosenandsincenoncausalsignalscan be shifted to the right,
in mostcaseswe canfind suchthat the signalDTFT of length canbe usedto
approximatewell the signalDTFT (and the signalFourier transform).
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Our goal is to numericallycomputetheDTFT (andtheFouriertransform),which
is a functionof thedigital frequency . Since is a -periodicfunction in
numericalcomputationsof thecorrespondingsumwe will needonly to considerthe
interval . For thepurposeof suchcomputations,this frequencyinterval
hasto bedivided into subintervalssothat thedigital frequenciesareevaluatedat
)
Notethatthesefrequenciesareequidistantlydistributedontheunit circle*+
. Now,
we are readyto definethe DFT, more preciselythe -point DFT of a length
signal as
),!-/.0132
- * 0 +54 ,!-6.01#2
- * 087:9 4;
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Thenumberof time samples andthenumberof thefrequencysamples can
be independentlychosen. It is convenientto make . If we can
add zerosat the end of the signal time samplesin order to increaseits length to
. This procedureis calledzero padding. It is demonstratedon the next example.
Note that the zeropaddingproceduredoesnot effect the DFT result.
Example 9.11: Considerthe following signalsamplesandassume
< = > ? @
This signal is zeropaddedto the lengthof as follows
< = > ? @
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Matrix Form of DFT
The -point DFT of thelength signalformulacanbeeasilyandconveniently
recordedusinga matrix form. Let us first form the vectorsof signal time samples
and the signal frequencydomainDFT samplesas
... ...
A B... C D B
The DFT formula canbe representedin the following matrix form
where the matrix is definedbyC EGF H�I D�J IKML D�J IONQP LR HSIC
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Observefirst that T�U T , and VXW T , which
implies thatall elementsin thefirst raw andin thefirst columnof thematrix are
equalto one. Simplealgebra(evaluationof Y Z U8[:\^]_ for given valuesof )
producesother entriesin matrix . For example
`�a5` bcaGb
Matrix Form for the Inverse of DFT (IDFT)
Assuming that matrixd aGe
is square ( ) and invertible, then
provides a simple formula for the recovery of the
signal time domain samplesfrom its frequency domain DFT samples. From
, under the above assumptions,we have the definition
of the inverseDFT transform, denotedby IDFT
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f6g
Thecrucialstepis to showthat thesquarematrix is invertible. This canbedone
by multiplying the matrix by its complexconjugate, h , obtainedby taking the
complexconjugateof eachelementin the matrix. It canbe easilyshownthat
h i
where i is anidentity matrix of dimension . Theaboveimpliesthat the inverse
of matrix hasa very simple form given by
f6g h
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Hence,the IDFT is givenby a very simpleformula
j6k l
Scalar Form of IDFT
From the last formula we can recoverevery single componentof the discrete-
time signal by observingthe following fact
j6k l lm�n o n8p:q^rs j mSnt
For eachparticularsamplevalue of , we havet j6kmSu3v
j m�nt mt j6km�u#v
o nw r m
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Hence,the scalar form of the IDFT is given byx�y/z{S|3} { ~&���5�
Note that the last formula was derived under the assumptionthat . If
that was not the case,then time domainsamplesof the vector would not be
uniquelyrecoveredfrom frequencydomainDFT samplesof the vector .
Discrete-Time Signal Wrapping (Modulo-N Reduction)
In the casewhen , a very simpleprocedurecalled the signalwrapping
can reducethe original signal length to . Note that the value for is dictated
by the conditionof the samplingtheorem,which sometimesproduceslarge values
for . The wrappingprocedureis demonstratedin the next example.
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Example 9.12: Let and with the signalvaluesgiven by
� � � � � � �
The modulo- reducedsignal (wrappedsignal) is given by
� � � � � � � �
In the casewhen is not an integermultiple of , the signalof length canbe
first zero paddedand then wrappedas demonstratedbelow.
Let and . We first form a signal of the length by
using the zero paddingprocedure,that is
� � � � � � � � �
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and thenwrap this signal into a signalof length as follows
� � � � � � � � � �
The wrappingprocedureis a consequenceof the following simple fact. Let us
denotethe wrappedsignal by , and let . Then the -point DFT of a
length signal is given by��� � ����� �6� � ���G� ���G� ���G������ �¢¡¤£
�6� �
��� � ���G� ��� �
where the columnsof matrix���G�
are identical to the first columnsof
matrix , that is
¥§¦ ¥�¦ ¨�© ¦OªQ«^¬ ¥�¦�
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It shouldbe emphasizedthat the inverseDFT for the wrappedsignal produces
the samplesof the wrappedsignal, that is
®6¯ °
IDFT in Terms of DFT
An interesting relations can be obtained from and
®6¯ ¯± °. It follows from that
° ° ° ° °
so we have
° °
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9.4.1 Fast Fourier Transform (FFT)
In order to calculatenumerically the DFT we have to multiply the
dimensionalmatrix by the vector of length . In general,multiplying a square
matrix of order by a correspondingvector requires ² scalarmultiplications.
In a celebratedpaperthat marks the beginningof the scientific discipline called
digital signal processing(Cooley and Tukey, 1965), it was shownhow to exploit
thespecialstructureof thematrix in orderto evaluatethe requiredproductmore
efficiently andto reducethe numberof the requiredscalarmultiplications. During
the last thirty five yearsmany the FFT algorithmsweredeveloped.The main idea
of thosealgorithmsis to evaluatethe –point DFT in termsof two –point
DFTs, andthento evaluatethe –point DFT in termsof two –point DFTs
and so on. Thosealgorithmshavea commonfeaturethat the requirednumberof
scalarmultiplicationsneededto evaluatethe DFT is given by
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³ ³
which in the caseof large valuesfor brings significantsavings. The function
³ growsmuchsloweras increasesthana linear function .
Detailed study of the FFT algorithms is beyond the scopeof this textbook.
Furthermore,sincedigital computersnow performscalarmultiplicationsandscalar
additionswith equalspeed,we will certainlyseein the nearfuture manynew FFT
algorithmsthatareefficient in view of boththenumberof bothscalarmultiplications
and scalaradditionsneededto evaluatethe DFT.
For thepurposeof thiscourse,it is sufficient to evaluatetheFFT usingMATLAB
and its function fft. This can be done in a very simple manneras follows
X=fft(x,N), where is the –datapoint vector and is the soughtDFT.
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Note that in evaluatingfft(x,N) the datavector is paddedwith zerosif it has
less than samplesand truncatedif hasmore than samples.The inverse
FFT is computedvia MATLAB using the function ifft asx=ifft(X,N).
9.5 Discrete-Time Fourier Series (DFS)
In Section9.1 we have introducedthe DTFT through the samplingoperationof
a continuous-timesignal and in Section 9.4 we have introducedthe DFT from
the DTFT. The DTFT could have been derived from the discrete-timeFourier
series(DFS) similarly to the Fourier transformbeing derived in Chapter3 from
the continuous-timeFourier series. Since we have alreadyintroducedthe DFT,
now the DFS comesas a by-productof DFT.
Discrete-timeFourierseriescanbe easilydeducedfrom the –point DFT and
its IDFT as the following pair
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´�µ/¶·S¸3¹ ¹ º¼» ·O½!¾ ¹
and
¹´�µ6¶
» ¸¿¹µ º» ·§½5¾ ¹
Knowing the fact that is a periodic signal with the period equal to , that
is , we concludethat the first formula representsthe periodic
signal expansionin terms of harmonicsof its fundamentaldigital harmonic
¹ —the discrete-timeFourier series(DFS). The fact that we needonly
harmonicto representa periodic signal follows from the result
ºÁÀ»&Ã¿Ä ´ÆÅ ·§½5¾ º» ·X½5¾ º�Ä ·�ÇÉÈ º&» ·X½5¾
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Consequently,the secondformula definesthe discrete-timeFourier seriescoeffi-
cients. Sincea periodicsignalhasan infinite time duration,thesignalobtainedvia
thediscrete-timeFourier serieshasto beperiodicallyextendedalongtheentire time
axis so that . The sameextensionshouldbe
donefor the negativevaluesof the discrete-timeinstants.
It canbe observedthat in contrastto the continuous-timeFourier series, which
representsa continuous-timeperiodic signal in terms of an infinite sum of its
harmonics, the DFS is a finite sumthat contains terms,where is the period
of the discrete-timeperiodic signal. Being representedby a finite sum, the DFS
is always convergent, in contrast to the continuous-timeFourier series whose
convergencerequiresthat the Dirichlet conditionsbe satisfied.
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Example 9.13: Considerthe following periodicdiscrete-timesignal
with theperiodequalto . The signalfundamentaldigital harmonicis equal
to Ê . The DFS coefficientsaregiven by
ÊË�Ì6ÍÎÏ Ê
Ì�Ð Î�ÑXÒ5Ó ÔÎÏ Ê
Ì�Ð Î�Ñ Õ Ö
Evaluatingeachof the four Fourier seriescoefficients,we obtain
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×�Ø Ù Ú ×�Ø�Û ×�Ø�Ü Ù Ú×�Ø�Û ×�Ø&ÝÉÛ ×�Ø�ÜÞÛ
×�Ø�Ü Ù Ú ×�Ø�Ü�Û ×�Ø�ß Ù Ú
Hence,the DFS representationof the given periodic signal isÜ
à�áãâ â ؼä à Ù Ú
Of course,this hasto be periodicallyextendedto any discrete-timeinstant since
. We can easily check that the DFS obtainedrepresentsthe
original signal, that is
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å æ ç å�è åé æ ç
å�è åêÉè å�é�è
å�é æ ç å�é�è å�ë æ ç
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Discrete-Time Linear System Response to Periodic Inputs
Similarly, as discussedin Section 5.4.1, we conclude that a hypotheti-
cal system input ìíOî!ï ðproducesthe discrete-timesystem output equal to
ì�í§î5ï ð ñ, where
ñis the discrete-timesystem digital fre-
quencytransferfunction. Using the linearity principle, the periodic input of the
period equal to and the fundamentaldigital harmonicequaltoñ
,
producesthe following zero-stateresponseòôó6õ
í§öñ ñ ñ ì ð íXî5ï
ò�ó6õ
í�öñ ñ ì ð í§î5ï
ñ ñ ñ
The systemoutput is also periodic with the sameperiod (the samefundamental
digital frequency)as the input signal.
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