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4.1 Chapter 4: Discrete-time Fourier Transform (DTFT)

4.1 DTFT and its Inverse

Forward DTFT: The DTFT is a transformation that maps Discrete-time (DT) signal x[n] into a complex valued

function of the real variable w, namely: wenxwXnjwn ,][)( (4.1)

· Note nis a discrete-time instant, but w represent the continuous real-valued frequency as in the

continuous Fourier transform. This is also known as the analysis equation.

· In general CwXÎ)(

· },{)()2(ppp-ÎÞ=+wwXnwX is sufficient to describe everything. (4.2) · )(wX is normally called the spectrum of ][nx with: ,:)(|:)(|.|)(|)()( (4.3) · The magnitude spectrum is almost all the time expressed in decibels (dB): |)(|log.20|)(|10wXwXdB= (4.4)

Inverse DTFT: Let )(wX be the DTFT of ].[nx Then its inverse is inverse Fourier integral of )(wX in the

interval ).,{pp- =-p p pdwewXnxjwn)(21][ (4.5)

This is also called the synthesis equation.

Derivation: Utilizing a special integral: ][2ndwejwnpdp p =ò- we write:

4.2 ][.2][][2][}][{)(][nxknkxdwekxdweekxdwewXkkknjwjwn

kjwkjwnpdpp pp pp p Note that since x[n] can be recovered uniquely from its DTFT, they form Fourier Pair: ).(][wXnxÛ Convergence of DTFT: In order DTFT to exist, the series å¥ njwn enx][ must converge. In other words: M Mnjwn MenxwX][)( must converge to a limit )(wX as .¥®M (4.6) Convergence of )(wXm for three difference signal types have to be studied: · Absolutely summable signals: ][nx is absolutely summable iff ¥å<¥ -¥=nnx|][|. In this case, )(wX always exists because: nnjwn njwn nxenxenx|][|||.|][||][| (4.7) · Energy signals: Remember ][nx is an energy signal iff .|][|2¥<庥 -¥=nxnxE We can show that )(wXM converges in the mean-square sense to :)(wX

0|)()(|2=ò--

¥®dwwXwXLimMMp

p (4.8) Note that mean-square sense convergence is weaker than the uniform (always) convergence of (4.7). · Power signals: ][nx is a power signal iff ¥<å+=-=¥®N

NnNxnxNLimP2|][|121

· In this case, ][nx with a finite power is expected to have infinite energy. But )(wXM may still converge

to )(wX and have DTFT. · Examples with DTFT are: periodic signals and unit step-functions. · )(wX typically contains continuous delta functions in the variable .w

4.3 4.2 DTFT Examples

Example 4.1 Find the DTFT of a unit-sample ].[][nnxd=

1][][)(0==å=å=-¥

-¥=-j njwn njwn eenenxwXd (4.9) Similarly, the DTFT of a generic unit-sample is given by:

0][]}[{00jwn

njwn eennnnDTFT-¥ =å-=-dd (4.10)

Example 4.2 Find the DTFT of an arbitrary finite duration discrete pulse signal in the interval: :21NN<

][][2 1 kncnxN

Nkk-å=-=d

Note: ][nx is absolutely summable and DTFT exists: jwkN

Nkknjwn

N

Nkknjwn

N

NkkecekncekncwX-

-=å=å-å=å-å=2 12 12 1 }][{]}[{)(dd (4.11)

Example 4.3 Find the DTFT of an exponential sequence: .1||][][<=awherenuanxn It is not difficult to see

that this signal is absolutely summable and the DTFT must exist. jw nnjw njwnn njwnnaeaeeaenuawX-¥ -=å=å=å=11 )(.][.)(00 (4.12) Observe the plot of the magnitude spectrum for DTFT and )(wXM for: 8.0=a and },20,10,5,2{DTFTM=¥=

4.4 Example 4.4 Gibbs Phenomenon: Significance of the finite size of M in (4.6).

For small

M, the approximation of a pulse by a finite harmonics have significant overshoots and undershoots. But it gets smaller as the number of terms in the summation increases.

Example 4.5 Ideal Low-Pass Filter (LPF). Consider a frequency response defined by a DTFT with a form:

<<<=pwwwwwX CC

0||1)( (4.13)

4.5 Here any signal with frequency components smaller than

Cw will be untouched, whereas all other frequencies will be forced to zero. Hence, a discrete-time continuous frequency ideal LPF configuration. Through the computation of inverse DTFT we obtain: )(21][pppnwSincwdwenxCCw wjwn C C (4.14) where . )sin()(xx xSincpp= The spectrum and its inverse transform for 2/p=C w has been depicted above.

4.3 Properties of DTFT

4.3.1 Real and Imaginary Parts:

][][][njxnxnxIR+= Û )()()(wjXwXwXIR+= (4.15)

4.3.2 Even and Odd Parts:

][][][nxnxnxoddev+= Û )()()(wXwXwXoddev+= (4.16a) ][]}[][.{2/1][**nxnxnxnxevev-=-+= Û ][]}[)(.{2/1)(**wXwXwXwXevev-=-+= (4.16b) ][]}[][.{2/1][**nxnxnxnxoddodd--=--= Û)()}()(.{2/1)(**wXwXwXwXoddodd--=--= (4.16c)

4.3.3 Real and Imaginary Signals:

If ÂÎ][nx then );()(*-=XwX even symmetry and it implies:

4.6 )()(|;)(|)(|wXwXwXwX--Ð=Ð-= (4.17a)

)()();()(wXwXwXwXIIRR--=-= (4.17b) If ÁÎ][nx (purely imaginary) then )()(*wXwX--=; odd symmetry (anti-symmetry.)

4.3.4 Linearity:

a. Zero-in zero-out and b. Superposition principle applies: )(.)(.][.][.wXBwXAnyBnxA+Û+ (4.18)

4.3.5 Time-Shift (Delay) Property:

)(.][wXeDnxjwD-Û- (4.19)

4.3.6 Frequency-Shift (Modulation) Property:

][.][nxewwXnjw

CC-Û- (4.20)

Example 4.6 Consider a first-order system:

]1[.][.][10-+=nxKnxKny Then )().()(10wXeKKwYjw-+= and the frequency response: jweKKwXwYjwH-+==.)(/)()(10

4.3.7 Convolution Property:

)().(][*][wHwXnhnxÛ (4.21)

4.3.8 Multiplication Property:

-Û-p p fffpdwYXnynx)().(21][].[ (4.22)

4.3.9 Differentiation in Frequency:

][.)(.nxndwwdXjÛ (4.23) 4.7

4.3.10 Parseval's and Plancherel's Theorems:

dwwXnxnòå=-¥ -¥=p p p22|)(|21|][| (4.24)

If][nx and/or ][ny complex then

dwwYwXnynxnò=å-¥ -¥=p p p)().(21][].[** (4.25) Example 4.7 Find the DTFT of a generic discrete-time periodic sequence ].[nx Let us write the Fourier series expansion of a generic periodic signal: =1 00 ][N knjkw keanx where Nwp2 0= )(2.){.){]}[{)(1 0 01 01

000å-=å=å==-

=N k knjkwN k kN knjkw kkwwaeDTFTaeaDTFTnxDTFTwXpd (4.26) Therefore, DTFT of a periodic sequence is a set of delta functions placed at multiples of

0kw with heights .ka

4.4 DTFT Analysis of Discrete LTI Systems

The input-output relationship of an LTI system is governed by a convolution process: ][*][][nhnxny= where ][nh is the discrete time impulse response of the system. Then the frequency-response is simply the DTFT of :][nh njwn wenhwH,].[)( (4.27)

4.8 · If the LTI system is stable then ][nh must be absolutely summable and DTFT exists and is continuous.

· We can recover ][nh from the inverse DTFT: ò ==-p p pdwewHwHIDTFTnhjwn).(21)}({][ (4.28) · We call |)(|wH as the magnitude response and )(wHÐ the phase response

Example 4.8 Let

][.)21 (][nunhn= and ][.)31 (][nunxn=

Let us find the output from this system.

1. Via Convolution: å

kknkknukunhnxny][.)21 ].([.)31 (][*][][ ÞNot so easy.

2. Via Fast Convolution or DFTF from Example 4.3 or Equation(4.12): jw

ewH--=21 11 )( and jw ewX--=31 11 )( jwjwjwjweeeewHwXwY------ --==31 1221
13 )21

1).(31

1(1 and the inverse DTFT will result in: ][.)31 (2][.)21 (3][nununynn-=

Example 4.9 Causal moving average system:

=1 0 ][1][M k knxMny

If the input were a unit-impulse: ][][nnxd= then the output would be the discrete-time impulse response:

4.9 ])[][(1

00/1][1][1

0

MnnnuMOtherwiseMnMknMnhM

k d

The frequency response:

)2/sin()2/sin(..11

1111)(2/)1(

2/2/2/2/

2/2/1

0wwMeMeeee

ee M ee

MeMwHMjw

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