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

1 Chapter 4: Discrete-time Fourier Transform (DTFT) 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,][)( ( ) 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 CwX )( },{)()2( =+wwXnwX is sufficient to describe everything. ( ) )(wX is normally called the spectrum of ][nx with: = angleSpectrumPhasewXSpectrumMagnitudewXe wXwXwXj,:)(|:)(|.|)(|)()( ( ) The magnitude spectrum is almost all the time expressed in decibels (dB): |)(| |)(|10wXwXdB= ( ) Inverse DTFT: Let )(wX be the DTFT of ].

2 [nx Then its inverse is inverse Fourier integral of )(wX in the interval ).,{ = dwewXnxjwn)(21][ ( ) This is also called the synthesis equation. Derivation: Utilizing a special integral: ][2ndwejwn = we write: ][.2][][2][}][{)(][nxknkxdwekxdweekxdwew Xkkknjwjwnkjwkjwn = = = = = = = Note that since x[n] can be recovered uniquely from its DTFT, they form Fourier Pair: ).(][wXnx Convergence of DTFT: In order DTFT to exist, the series = njwnenx][ must converge. In other words: = = MMnjwnMenxwX][)( must converge to a limit )(wX as . M ( ) Convergence of )(wXm for three difference signal types have to be studied: Absolutely summable signals: ][nx is absolutely summable iff < =nnx|][|.]}

3 In this case, )(wX always exists because: < = = = = nnjwnnjwnnxenxenx|][|||.|][||][| ( ) Energy signals: Remember ][nx is an energy signal iff .|][|2 < =nxnxE We can show that )(wXM converges in the mean-square sense to :)(wX 0|)()(|2= dwwXwXLimMM ( ) Note that mean-square sense convergence is weaker than the uniform (always) convergence of ( ). Power signals: ][nx is a power signal iff < += = NNnNxnxNLimP2|][|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 DTFT Examples Example Find the DTFT of a unit-sample ].

4 [][nnx = 1][][)(0== = = = = jnjwnnjwneenenxwX ( ) Similarly, the DTFT of a generic unit-sample is given by: 0][]}[{00jwnnjwneennnnDTFT = = = ( ) Example Find the DTFT of an arbitrary finite duration discrete pulse signal in the interval: :21NN< ][][21kncnxNNkk = = Note: ][nx is absolutely summable and DTFT exists: jwkNNkknjwnNNkknjwnNNkkecekncekncwX = = = = = = = =212121}][{]}[{)( ( ) Example 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. jwnnjwnjwnnnjwnnaeaeeaenuawX = = = = = = =11)(.][.)(00 ( ) Observe the plot of the magnitude spectrum for DTFT and )(wXM for: and },20,10,5,2{DTFTM= = Example Gibbs Phenomenon: Significance of the finite size of M in ( ).]

5 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 Ideal Low-Pass Filter (LPF). Consider a frequency response defined by a DTFT with a form: <<<= wwwwwXCC0||1)( ( ) 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][ nwSincwdwenxCCwwjwnCC= = ( ) where .)sin()(xxxSinc = The spectrum and its inverse Transform for 2/ =Cw has been depicted above. Properties of DTFT Real and Imaginary Parts: ][][][njxnxnxIR+= )()()(wjXwXwXIR+= ( ) Even and Odd Parts: ][][][nxnxnxoddev+= )()()(wXwXwXoddev+= ( ) ][]}[][.

6 {2/1][**nxnxnxnxevev = += ][]}[)(.{2/1)(**wXwXwXwXevev = += ( ) ][]}[][.{2/1][**nxnxnxnxoddodd = = )()}()(.{2/1)(**wXwXwXwXoddodd = = ( ) Real and Imaginary Signals: If ][nx then );()(* =XwX even symmetry and it implies: )()(|;)(|)(|wXwXwXwX = = ( ) )()();()(wXwXwXwXIIRR = = ( ) If ][nx (purely imaginary) then )()(*wXwX =; odd symmetry (anti-symmetry.) Linearity: a. Zero-in zero-out and b. Superposition principle applies: )(.)(.][.][.wXBwXAnyBnxA+ + ( ) Time-Shift (Delay) Property: )(.][wXeDnxjwD ( ) Frequency-Shift (Modulation) Property: ][.][nxewwXnjwCC ( ) Example Consider a first-order system: ]1[.][.][10 +=nxKnxKny Then )().}

7 ()(10wXeKKwYjw += and the frequency response: jweKKwXwYjwH +==.)(/)()(10 Convolution Property: )().(][*][wHwXnhnx ( ) Multiplication Property: dwYXnynx)().(21][].[ ( ) Differentiation in Frequency: ][.)(.nxndwwdXj ( ) Parseval s and Plancherel s Theorems: dwwXnxn = = 22|)(|21|][| ( ) If][nx and/or ][ny complex then dwwYwXnynxn = = )().(21][].[** ( ) Example Find the DTFT of a generic Discrete-time periodic sequence ].[nx Let us write the Fourier series expansion of a generic periodic signal: = =100][Nknjkwkeanx where Nw 20= )(2.){.){]}[{)(100101000 = = == = = =NkknjkwNkkNknjkwkkwwaeDTFTaeaDTFTnxDTFT wX ( ) Therefore, DTFT of a periodic sequence is a set of delta functions placed at multiples of 0kw with heights.}}

8 Ka 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 = = njwnwenhwH,].[)( ( ) 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: == dwewHwHIDTFT nhjwn).(21)}({][ ( ) We call |)(|wH as the magnitude response and )(wH the phase response Example Let ][.)21(][nunhn= and ][.)31(][nunxn= Let us find the output from this system. 1. Via Convolution: == = kknkknukunhnxny][.)21].([.)31(][*][][ Not so easy.

9 2. Via Fast Convolution or DFTF from Example or Equation( ): jwewH =2111)( and jwewX =3111)( jwjwjwjweeeewHwXwY = ==31122113)211).(311(1)().()( and the inverse DTFT will result in: ][.)31(2][.)21(3][nununynn = Example Causal moving average system: = =10][1][MkknxMny If the input were a unit-impulse: ][][nnx = then the output would be the Discrete-time impulse response: ])[][(100/1][1][10 MnnnuMOtherwiseMnMknMnhMk =< = = = The frequency response: )2/sin()2/sin(..111111)(2/)1(2/2/2/2/2/2 /10wwMeMeeeeeeMeeMeMwHMjwjwjwjwMjwMjwjwM jwjwMMnjwn = = = = = For M=6 we plot the magnitude and the phase response of this system: Notes: 1. Magnitude response Zeros at 0)/sin()2/sin(2==wwwMwhereMkw 2. Level of first sidelobe dB13 3.

10 Phase response with a negative slope of 2/)1( M 4. Jumps of at )/sin()2/sin(2wwwMwhereMkw =changes its sign. TABLE: Discrete-time Fourier Transform PAIRS Signal DTFT ][n 1 1 )(.2w njwCe )(.2 Cww =CNknjkwkNwwitheaC =10)(..2 NkCkkwwa 1||];[.<nnuan jwea .11 1||.;||<nan + 1||];[..<nnuann 2).1(.jwjweaea ][Nnrect ]2/sin[)]2/1(sin[wNw+ nnwC sin ]2/[Cwwrect TABLE PROPERTIES OF DTFT 1. Linearity ][.][.21nxBnxA+ )(.)(.21wXBwXA+ 2. Time-Shift (Delay) ][Nnx )(.wXejwN 3. Frequency-Shift njwCenx].[ )(CwwX 4. Linear Convolution ][*][nhnx )().(wHwX 5. Modulation ][].[npnx > < 2)().(21dwHX 6. Periodic Signals ][][Nnxnx+= =kCkkwwa)(..2 NwC 2= => < NnjkwkCenxNa].[1 Example Response of an LTI system with :]}[{)(nhDTFTwH= Given jwnenx=][; a complex harmonic.