DSP-I DSP-I DSP-I DSP-I

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1 NOTES FOR 8-79 LECTURES 3 and 4 Introduction to Discrete-Time Fourier Transforms (DTFTs Distributed: September 8, 2005 Notes: This handout contains in brief outline form the lecture notes used for 8-79 lectures Number 3 and 4, presented by videotape on September 6 and 8. Because the notes were transcribed some time after the lecture was taped, there may be some minor differences between these notes and what is seen on the video. Also, topics have been merged across the two lectures, so the order of presentation may vary somwhat from what is in the lecture. I. Introduction Last week we talked about the time domain behavior of discrete-time signals and systems. Today we will begin (or more accurately we will begin to review how we characterize discrete-time signals and systems in the frequency domain using the discrete-time Fourier transform. II. The DTFT and its inverse Digital Signal Processing I (8-79 Fall Semester, 2005 Probably the easiest way to introduce the discrete-time Fourier transform (DTFT is through its counterpart, the continuous-time Fourier transform (CTFT. As you will recall, the CTFT and its inverse can be expressed as: xt ( XjΩ ( e jωt dt ( XjΩ ( xt (e jωt dt (2 Note that we use the upper case variable Ω to indicate continuous-time angular frequency in this course. As we have discused, the first equation (which is actually the inverse CTFT expresses the fact that a finiteenergy time function xt ( can be represented as a weighted linear combination of complex exponentials e jωt. The second equation (the actual CTFT tells you how to compute the weights XjΩ (. Note that the frequencies Ω include all real numbers. The DTFT Xe jω ( is computed very similar fashion to the CTFT:

2 8-79 Lecture 3 Notes -2- Fall, 2005 Xe ( jωn xn [ ]e jωn (3 Note that the complex exponential e jωn in Eq. 3 has the same role as the complex exponential e jωt in Eq.. Now let s consider the expression e jωn for the frequencies ω 0 and ω 0 + : e j ( ω 0 + n n e jn n (4 In other words, the discrete-time frequency variable is periodic with period. As a result, we only evaluate the IDTFT computation over a frequency region that is arbitrary but of extent : This periodicity occurs because the time variable n is always integer. xn [ ] Xe ( jω e jωn dω where the single subscript on the integral sign indicates that integration can be performed over any strip of ω of extent. This occurs because the entire integrand of Eq. 5 is periodic. The validity of Eqs. (3 and (5 can be confirmed by simple substitution: (5 xn [ ] Xe ( jω e jωn dω xl [] l e jωl e jωn dω (6 xl []----- e jωl e jωn dω l l (7 Because n and l are always integers, the integral in Eq. (7 is equal to when n l and zero otherwise. Hence the only nonzero term in the outer sum occurs when l n, and it evaluates to xn [ ]. This confirms that Eqs. (3 and (5 are transform pairs. III. Basic DTFT examples The decaying exponential. We first consider the simple function xn [ ] α n un [ ], 0 < n < Substituting directly produces Xe ( jω xn [ ]e jωn α n e jωn ( αe jω n n n 0 n αe jω (8 (9 As shown in class, the real and imaginary parts can be obtained by rationalizing the denominator: Xe jω ( (0 αe jω αe jω αe jω αe jω αcos( ω jαsin( ω αe jω

3 8-79 Lecture 3 Notes -3- Fall, 2005 By comparison of the terms we obtain Re[ Xe ( jω ] αcos( ω ( and Im[ Xe ( jω ] jαsin( ω (2 Because Z 2 ( Re[ Z] 2 + ( Im[ Z] 2 Im[ Z] and Z atan for any complex variable Z, Re[ Z] Xe ( jω 2 αcos( ω αsin( ω ( 2 (3 Hence Xe jω ( (4 and Xe ( jω atan αsin( ω αcos( ω (5 We note that Re[ Xe ( jω ] is an even function of ω, Im[ Xe ( jω ] is odd, Xe ( jω is even and Xe ( jω is odd, at least for this particular time function. E2. The finite-duration pulse. We next consider the finite duration causal pulse, xn [ ] 0, n N 0, otherwise This sum is also easily obtained: Xe ( jω xn [ ]e jωn N e jωn N ( e jω n n n 0 n 0 e jωn e jω (6 (7 Please note that we used the relation for the finite sum of exponentials discussed in recition in developing the last result. This expression can be further simplified by balancing the terms in the parentheses: Xe jω ( e jωn e jωn 2 jωn 2 e e jωn e jω e jω e jω 2 e jω 2 e jω( N 2 2j Nω sin( 2jsin( ω 2 (8 or

4 8-79 Lecture 3 Notes -4- Fall, 2005 Xe ( jω e jω( N sin( Nω 2 sin( ω 2 (9 The term on the left contributes a linear phase shift to the DTFT. The quantity inside the parentheses, , is sometimes referred to as the discrete-time sinc function. It has the following proper- sin( Nω 2 sin( ω 2 ties: sin( Nω 2 By L Hopital s rule, lim ω 0 sin( ω 2 N sin( Nω 2 has regularly-occurring zero crossings at for all integer k sin( ω 2 ω k The envelope of sin( Nω 2 tapers downward as increases from zero to sin( ω 2 ω π IV. Some properties and additional examples of DTFTs In this section we summarize some of the properties of DTFTs along with some additional DTFT examples. In some cases, brief proofs were provided in the lecture... for the most part these proofs are not included in these notes in the interests of brevity. P. Linearity. As noted in class, the DTFT operation itself is linear. P2. Time shift. xn [ N] Xe ( jω e jωn P3. Multiplication by a complex exponential. xn [ ] n X e j ( ω ω 0 E3. DTFT of an impulse. δ[ n] δ[ n]e jωn for all n. n E4. DTFT of a constant. In similar fashion, working from the right side we obtain δ( ω r r (20 While this notation is cumbersome, it merely expresses the fact that a constant in time has a DTFT that is a delta function in frequency at ω 0 and that this function repreats periodically in frequency with period. E5. DTFT of a complex exponential. Using Property P3, we can now easily obtain n δ( ω ω 0 r. (2 r

5 8-79 Lecture 3 Notes -5- Fall, 2005 This is simply a shifted delta function that is repeated periodically. E6. DTFT of a cosine. From Euler s representation of trig functions we obtain cos( ω 0 n + n π δ( ω ω 2 ( 0 r + δω ( + ω 0 r r (22 This is simply a pair of delta functions of area π occuring at ω ± ω 0, repeating periodically with period. P4. Time reversal. x[ n] Xe ( jω We note that if a time function is even, xn [ ] x[ n] Xe ( jω Xe ( jω Similarly, if a time function is odd, xn [ ] x[ n] Xe ( jω X' ( e jω In other words, if a time function is even, its DTFT is also even, and if a time function is odd its DTFT is also odd. P5. Complex conjugation. x [ n] X ( e jω We note that if a time function is real, xn [ ] x [ n] Xe ( jω X ( e jω. This latter property is referred to as Hermitian symmetry. If Xe ( jω is Hermetian symmetric, its real part is even, its imaginary part is odd, its magnitude is real, and its phase angle is odd. We note further that if a real time function is even or odd, both the Hermitian symmetry and the time reversal properties constrain the transform. Specifically, if a time function is both real and even, its DTFT is also real and even. If a time function is real and odd, its DTFT is both imaginary and odd. For example, the sine wave has the transform sin( ω 0 n n π δω ( ω 2j j ( 0 r δω ( + ω 0 r r (23 P6. Parseval s theorem. xn [ ] 2 n Xe ( jω 2 dω (24 This relationship is important because the quantity on the left side of the equation is clearly the total energy of the time function. Hence the energy in a frequency band can be obtained by integrating the function Xe ( jω 2 over that frequency band. (Be sure to take the negative frequencies into account! Because of its phsycial meanining, the function Xe jω ( 2 is sometimes referred to as the energy density spectrum. P7. Initial value theorems. These trivial-to-prove theorems are sometimes confused with Parseval s theorem. They sometimes help with computation. n xn [ ] Xe ( jω ω 0 (25

6 8-79 Lecture 3 Notes -6- Fall, 2005 x[ 0] Xe ( jω dω (26