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[PDF] Chapter 3: Problem Solutions

Using the definition determine the DTFT of the following sequences It it does not exist Solution a) DFT x 3 , x 0 , x 1 , x 2 = DFT x n - 1 4 = w4 The inverse DCT obtained for L = 20, 30, 40 are shown below

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SP-3 45 Find the DTFT of the signal x(n) = -a" u-n – 1), a is real Solution Solution From the definition of inverse DTFT, we have x(n) = Il = ejln d2 X 27 -7

[PDF] Discrete-Time Fourier Transform - Higher Education Pearson

This example is actually strong enough to justify that the inverse DTFT The result given (7 48a) and (7 48b) is easily seen from a graphical solution that

[PDF] Chapter 4: Discrete-time Fourier Transform (DTFT) 41 DTFT and its

wX be the DTFT of ] [ nx Then its inverse is inverse Fourier integral of )( Examples with DTFT are: periodic signals and unit step-functions • )( wX typically 

[PDF] The Discrete Fourier Transform - Eecs Umich

Example Find N-point inverse DFT of {X[k]}N−1 k=0 where X[k] = { 1, k = k0 0, otherwise= In fact in this case there is an analytical solution: x[n] = 1 4 sinc3(1

12 Discrete Fourier transform

17 nov 2006 · Example 12 2 Calculate the inverse DFT of X[r] = 5 r = 0 3 − j2 r = 1 −3 r = 2 3 + j2 r = 3 Solution Using Eq (12 13), the 

[PDF] Problem set solution 11: Discrete-time Fourier transform

11 Discrete-Time Fourier Transform Solutions to Recommended Problems S11 1 (a) X(Q) = 7 x[n]e -jn" = (i" [nje -a = (le -j")n n=O 1 1 -ie- Here we have 

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DFT Sample Exam Problems with Solutions 1 Consider -point Discrete Fourier Transform (DFT) of ),( yxf is obtained by the Inverse DFT of the signal ),( ),(

[PDF] Topic 5: Discrete-Time Fourier Transform (DTFT) - RGIT

Figures and examples in these course slides are taken from the following sources: Transform Fourier DT Inverse ][ )( P CT DT : Transform Fourier DT deeX Solution ○ This can be solved using convolution of h[n] and x[n] ○ However 

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CHAPTER

7Discrete-TimeFourierTransform

In Chapter 3 and Appendix C, we showed that interesting continuous-time waveforms x(t)can be synthesized by summing sinusoids, or complex exponential signals, having different frequenciesf k and complex amplitudesa k . We also introduced the concept of thespectrumof a signal as the collection of information about the frequencies and correspondingcomplexamplitudes{f k ,a k }ofthecomplexexponentialsignals,andfound it convenient to display the spectrum as a plot of spectrum lines versus frequency, each labeled with amplitude and phase. This spectrum plot is afrequency-domain representationthat tells us at a glance “how much of each frequency is present in the signal." In Chapter 4, we extended the spectrum concept from continuous-time signalsx(t) to discrete-time signalsx[n]obtained by samplingx(t). In the discrete-time case, the line spectrum is plotted as a function of normalized frequency. In Chapter 6, we developed the frequency responseH(ej )which is the frequency-domain representation of an FIR lter. Since an FIR lter can also be characterized in the time domain by its impulse response signalh[n], it is not hard to imagine that the frequency response is the frequency-domain representation,orspectrum, of the sequenceh[n]. 236

7-1 DTFT: FOURIER TRANSFORM FOR DISCRETE-TIME SIGNALS237

(DTFT).The DTFT is a frequency-domain representation for a wide range of both finite- and infinite-length discrete-time signalsx[n]. The DTFT is denoted asX(e ), which shows that the frequency dependence always includes the complex exponential function e . The operation of taking the Fourier transform of a signal will become a common tool for analyzing signals and systemsin the frequency domain. 1 The application of the DTFT is usually called Fourier analysis, or spectrum analysis or “going into the Fourier domain or frequency domain." Thus, the words spectrum, Fourier, and frequency-domain representation become equivalent, even though each one retains its own distinct character.

7-1 DTFT: FourierTransform for Discrete-Time Signals

The concept of frequency response discussed in Chapter 6 emerged from analysis showing that if an input to an LTI discrete-time system is of the formx[n]=e then the corresponding output has the formy[n]=H(e )e , whereH(e )is called the frequency response of the LTI system. This fact, coupled with the principle of functionH(e )is sufficient to determine the output due to any linear combination of signals of the forme FIR filters discussed in Chapter 6, the frequency response function is obtained from the summation formula H(e M n=0 h[n]e =h[0]+h[1]e +···+h[M]e (7.1) whereh[n]is the impulse response. In a mathematical sense, the impulse responseh[n] istransformedinto the frequency response by the operation of evaluating (7.1) for each the time-domain representationh[n]is replaced by the frequency-domain representation H(e ). For this notion to be complete and useful, we need to know that the result of the transformation is unique, and we need the ability to go back from the frequency-domain representation to the time-domain representation. That is, we need aninverse transform that recovers the originalh[n]fromH(e ). In Chapter 6, we showed that the sequence can be reconstructed from a frequency response represented in terms of powers ofe as in (7.1) by simply picking off the coefficients of the polynomial since,h[n]is the coefficient ofe . While this process can be effective ifMis small, there is a much sequences. 1

It is common in engineering to say that we “take the discrete-time Fourier transform" when we mean

that we considerX(e )as our representation of a signalx[n].

238CHAPTER 7 DISCRETE-TIME FOURIER TRANSFORM

In this section, we show that the frequency response is identical to the result of applying the more general concept of the DTFT to the impulse response of the LTI system. We give an integral form for the inverse DTFT that can be used even when H(e jˆω )does not have a nite polynomial representation such as (7.1). Furthermore, we show that the DTFT can be used to represent a wide range of sequences, including sequences of innite length, and that these sequences can be impulse responses, inputs to LTI systems, outputs of LTI systems, or indeed, any sequence that satises certain conditions to be discussed in this chapter.

7-1.1 Forward DTFT

The DTFT of a sequencex[n]is dened as

Discrete-Time Fourier Transform

X(e jˆω n=-∞ x[n]e -jˆωn (7.2)

The DTFTX(e

jˆω )that results from the denition is a function of frequencyˆω. Going from the signalx[n]to its DTFT is referred to as ìtaking the forward transform,î and going from the DTFT back to the signal is referred to as ìtaking the inverse transform.î Thelimitsonthesumin (7.2)areshownasinnitesothattheDTFTdenedforinnitely long signals as well as nite-length signals. 2

However, a comparison of (7.2) to (7.1)

shows that if the sequence were a nite-length impulse response, then the DTFT of that sequencewouldbethesameasthefrequencyresponseoftheFIRsystem. Moregenerally, ifh[n]is the impulse response of an LTI system, then the DTFT ofh[n]is the frequency responseH(e jˆω )of that system. Examples of innite-duration impulse response lters will be given in Chapter 10.

EXERCISE 7.1Show that the DTFT functionX(e

jˆω )dened in (7.2) is always periodic inˆωwith period 2π, that is, X(e j(ˆω+2π) )=X(e jˆω

7-1.2 DTFT of a Shifted Impulse Sequence

OurrsttaskistodevelopexamplesoftheDTFTforsomecommonsignals. Thesimplest case is the time-shifted unit-impulse sequencex[n]=δ[n-n 0 ]. Its forward DTFT is by denition X(e jˆω n=-∞

δ[n-n

0 ]e -jˆωn 2 The innite limits are used to imply that the sum is over alln, wherex[n]?=0. This often avoids unnecessarily awkward expressions when using the DTFT for analysis.

7-1 DTFT: FOURIER TRANSFORM FOR DISCRETE-TIME SIGNALS239

Since the impulse sequence is nonzero only atn=n

0 it follows that the sum has only one nonzero term, so X(e jˆω )=e -jˆωn 0 To emphasize the importance of this and other DTFT relationships, we use the notation DTFT ←→to denote the forward and inverse transforms in one statement: 0 x[n]=δ[n-n 0 DTFT ←→X(e jˆω )=e -jˆωn 0 (7.3)

7-1.3 Linearity of the DTFT

most important properties. The DTFT is a linear operation; that is, the DTFT of a sum of two or more scaled signals results in the identical sum and scaling of their corresponding

DTFTs. To verify this, assume thatx[n]=ax

1 [n]+bx 2 [n], whereaandbare (possibly complex) constants. The DTFT ofx[n]is by definition X(e jˆω n=-∞ (ax 1 [n]+bx 2 [n])e -jˆωn

If bothx

1 [n]andx 2 [n]have DTFTs, then we can use the algebraic property that multiplication distributes over addition to write X(e jˆω )=a n=-∞ x 1 [n]e -jˆωn +b n=-∞ x 2 [n]e -jˆωn =aX 1 (e jˆω )+bX 2 (e jˆω That is, the frequency-domain representations are combined in exactly the same way as the signals are combined.

EXAMPLE 7-1 DTFT of an FIR Filter

The following FIR filter

y[n]=5x[n-1]-4x[n-3]+3x[n-5] has a finite-length impulse response signal: h[n]=5δ[n-1]-4δ[n-3]+3δ[n-5] Each impulse inh[n]is transformed using (7.3), and then combined according to the linearity property of the DTFT which gives H(e jˆω )=5e -jˆω -4e -j3ˆω +3e -j5ˆω

240CHAPTER 7 DISCRETE-TIME FOURIER TRANSFORM

7-1.4 Uniqueness of the DTFT

);inotherwords,twodifferent signals cannot have the same DTFT. This is a consequence of the linearity property because if two different signals have the same DTFT, then we can form a third signal by subtraction and obtain x 3 [n]=x 1 [n]Šx 2 [n] DTFT X 3 (e )=X 1 (e )ŠX 2 (e identical DTFTs =0 However, from the denition (7.2) it is easy to argue thatx 3 [n]has to be zero if its DTFT is zero, which in turn implies thatx 1 [n]=x 2 [n]. The importance of uniqueness is that if we know a DTFT representation such as (7.3), we can start in either the time or frequency domain and easily write down the corresponding representation in the other domain. For example, ifX(e )=e then we know thatx[n]=←[nŠ3].

7-1.5 DTFT of a Pulse

Another common signal is theL-point rectangular pulse, which is a nite-length time signal consisting of all ones: r L [n]=u[n]Šu[nŠL]=

1n=0,1,2,...,LŠ1

0 elsewhere

Its forward DTFT is by denition

R L (e

LŠ1

n=0 1e =1Še

1Še

(7.4) where we have used the formula for the sum ofLterms of a geometric series to ìsumî the series and obtain a closed-form expression forR L (e ). This is a signal that we studied before in Chapter 6 as the impulse response of anL-point running-sum lter. In Section 6-7, the frequency response of the running-sum lter was shown to be the product of a Dirichlet form and a complex exponential. Referring to the earlier results in Section 6-7 or further manipulating (7.4), we obtain another DTFT pair:

DTFT Representation ofL-Point Rectangular Pulse

r L [n]=u[n]Šu[nŠL] DTFT R L (e (7.5) Since the lter coefcients of the running-sum lter areLtimes the lter coefcients of the running-average lter, there is noLin the denominator of (7.5).

7-1 DTFT: FOURIER TRANSFORM FOR DISCRETE-TIME SIGNALS241

7-1.6 DTFT of a Right-Sided Exponential Sequence

As an illustration of the DTFT of an infinite-duration sequence, consider a “right-sided" exponential signal of the formx[n]=a n u[n], whereacan be real or complex. Such a signal is zero forn<0 (on theleft-hand sideof a plot). It decays “exponentially" for n≥0if|a|<1; it remains constant at 1 if|a|=1; and it grows exponentially if|a|>1.

Its DTFT is by definition

X(e jˆω n=-∞ a n u[n]e -jˆωn n=0 a n e -jˆωn

We can obtain a closed-form expression forX(e

jˆω )by noting that X(e jˆω n=0 (ae -jˆω n which can now be recognized as the sum of all the terms of an infinite geometric series, where the ratio between successive terms is(ae -jˆω ). For such a series there is a formula for the sum that we can apply to give the final result X(e jˆω n=0 (ae -jˆω n =1 1-ae -jˆω There is one limitation, however. Going from the infinite sum to the closed-form result is only valid when|ae -jˆω |<1or|a|<1. Otherwise, the terms in the geometric series grow without bound and their sum is infinite. ThisDTFTpairisanotherwidelyusedresult, worthyofhighlightingaswehavedone with the shifted impulse and pulse sequences.

DTFT Representation ofa

n u[n] x[n]=a n u[n] DTFT ←→X(e jˆω )=1 1-ae -jˆω if|a|<1 (7.6) X(e jˆω )=1

1-0.5e

-jˆω EXERCISE 7.3Use the linearity of the DTFT and (7.6) to determine the DTFT of the following sum of two right-sided exponential signals:x[n]=(0.8) n u[n]+2(-0.5) n u[n].

242CHAPTER 7 DISCRETE-TIME FOURIER TRANSFORM

7-1.7 Existence of the DTFT

In the case of nite-length sequences such as the impulse response of an FIR lter, the sum dening the DTFT has a nite number of terms. Thus, the DTFT of an FIR lter as in (7.1) always exists becauseX(e jˆω )is always nite. However, in the general case, where one or both of the limits on the sum in (7.2) are innite, the DTFT sum may diverge (become innite). This is illustrated by the right-sided exponential sequence in

Section 7-1.6 when|a|>1.

the following manipulation that develops a bound on the size ofX(e jˆω |X(e jˆω n=-∞ x[n]e -jˆωn n=-∞ x[n]e -jˆωn n=-∞ |x[n]| 1e -jˆωn (magnitude of product=product of magnitudes) n=-∞ |x[n]| It follows that a sufcient condition for the existence of the DTFT ofx[n]is

Sufficient Condition for Existence of the DTFTX(e

jˆω n=-∞ |x[n]|<∞ (7.7) Asequencex[n]satisfying(7.7)issaidtobeabsolutelysummable, andwhen (7.7)holds, the innite sum dening the DTFTX(e jˆω )in (7.2) is said toconvergeto a nite result for allˆω.

EXAMPLE 7-2 DTFT of Complex Exponential?

Consider a right-sided complex exponential sequence,x[n]=re jˆω 0 n u[n]whenr=1. Applying the condition of (7.7) to this sequence leads to n=0 |e jˆω 0 n n=0

1→∞

Thus, the DTFT of a right-sided complex exponential is not guaranteed to exist, and it is easy to verify that|X(e jˆω 0 )|→∞. On the other hand, ifr<1, the DTFT of x[n]=r n e jˆω 0 n u[n]exists and is given by the result of Section 7-1.6 witha=re jˆω 0 The non-existence of the DTFT is also true for the related case of a two-sided sinusoid, dened ase jˆω 0 n for-∞7-1 DTFT: FOURIER TRANSFORM FOR DISCRETE-TIME SIGNALS243

7-1.8 The Inverse DTFT

Now that we have a condition for the existence of the DTFT, we need to address the question of the inverse DTFT. The uniqueness property implies that if we have a table of known DTFT pairs such as (7.3), (7.5), and (7.6), we can always go back and forth between the time-domain and frequency-domain representations simply by table lookup as in Exercise 7.2. However, with this approach, we would always be limited by the size of our table of known DTFT pairs. Instead, we want to continue the development of the DTFT by studying a general expression for performing the inverse DTFT. The DTFTX(e jˆω )is a function of the continuous variableˆω, so an integral (7.8) with respect to normalized frequencyˆωis needed to transformX(e jˆω )back to the sequencex[n].

Inverse DTFT

x[n]=1 2π X(e jˆω )e jˆωn dˆω. (7.8) Observe thatnis an integer parameter in the integral, whileˆωnow is a dummy variable of integration that disappears when the definite integral is evaluated at its limits. The variablencan take on all integer values in the range-∞DTFT in (7.2) into (7.8) and rearranging terms. Instead of carrying out a general proof, we present a simpler and more intuitive justification by working with the shifted impulse sequenceδ[n-n 0 ], whose DTFT is known to bequotesdbs_dbs14.pdfusesText_20