8 The Discrete Fourier Transform (DFT)
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1 8 The Discrete Fourier Transform (DFT) ² Discrete-Time Fourier Transform and Z-transform are de ned over in niteduration sequence. Both transforms are functions of continuous variables (ω and z). For nite-duration sequences, one can de ne an alternative Fourier representation, called Discrete Fourier Transform, which is a function of discrete variables. In general, we represent DFT as a sequence. Consider complex sequences below. 8.1 Representation of Periodic Sequences: The Discrete Fourier Series ² Let a complex sequence ex [n] be periodic with period (i.e., ex [n] ex [n + r] for any integer values of n and r). Any complex periodic sequence of period can be represented by a linear sum of complex exponentials with frequencies being integer multiples of the fundamental frequency 2π, as ex [n] 1 1 k0 e [k] e j2π k n. ~ This expansion is called the Fourier series representation of ex [n] with coe cients e [k] s. j2π kn ote: Since there are di erent complex exponential sequences e, the summation extends over frequency variable k from 0 to 1. ² How can we obtain the coe cient e [k] from ex [n]? 1
2 ow, for an integer r, 1 1 k0 ex [n] e j2π r n 1 1 k0 1 e [k] Ã 1 e [r] ) e [k] 1 e [k] e j2π (k r)n 1 1 e j2π (k r)n {z } 1, if k r e j2π αn 1 e j2π α ex [n] e j2π n k. 0 for integer a e j2π α ote: e [k] is also periodic with period in general and thus can be represented by a Fourier series expansion. ² We shall de ne hereafter with 2π W, e j2π being the fundamental frequency and refer to the following! ex [n] 1 1 k0 e [k] W kn (Synthesis Equation) and e [k] 1 ex [n] W kn (Analysis Equation) as the Discrete Fourier Series (DFS) pair, denoted by ex [n] DFS Ã! e [k]. 2
3 ² Ex: DFS of a Periodic Impulse Train ex [n] 1 r 1 ) e 1 [k] ex [n] 1 δ [n r] δ [n] W kn 1 k0 W kn 1 for all k 1 Fig. 81-F1 ote that P 1 r 1 δ [n r] P 1 1 k0 e j2π nk. ² Ex: Duality in the DFS 1 k0 e j2π nk Consider the DFS coe cients as a periodic impulse train ey [k] ) ey [n] 1 1 r 1 1 ey [k] 1 k0 δ [k r] δ [k] W kn 1 W kn 1 for all n e j2π nk. ² Ex*: The DFS of a Periodic Rectangular Pulse Train Let be even and consider ½ 1, l n < l + ex [n] 2 0, l + ; 8l integer n < (l + 1) 2 ) e [k] /2 1 W kn 1 W k/2 1 W k Fig
4 ow, we further have ² ow, let e [k] W k/2 2 ³ W k 2 W k/2 2 ³ W e j2π W k(/2 1) 2 µ W k sin kπ W k/2 2 W k 2 W k 2 sin kπ 2 e j2π 2 2 W 2 k e j2π k 2 e jπk( ) sin kπ 2 sin. kπ x [n] and nd its DT Fourier transform Fig. 8.2 e j2π 2 W 2 2 e j2π k 2 ½ ex [n], 0 n 1 e jω 1 x [n] e jωn ) e [k] e jω ω 2π k. 1 µ πk 2j sin ex [n] e jωn Thus, the DFS e [k] is a sequence of equally-spaced samples of (e jω ), with a frequency spacing of 2π. ote: In Ex*, if we let ½ 1, x [n] ) e jω 0 n 2 1 0, elsewhere /2 1 ) e [k] e jω ω 2π k 8.2 Properties of DFS e jωn e j( 2 1)ω sin ω 4 sin ω 2 ² Assume that all sequences below have period. The DFS has the following properties: 4.
5 8.2.1 Linearity: If then ex 1 [n] DFS Ã! e 1 [k] ex 2 [n] DFS Ã! e 2 [k] aex 1 [n] + bex 2 [n] DFS Ã! a e 1 [k] + b e 2 [k]. (In general, superposition principle holds.) Shifting of a Sequence: If then ex [n] DFS Ã! e [k] ex [n m] 1 1 k0 e [k] W k(n m) ote: ex [n m 1 ] ex [n m 1 l], l integer. Furthermore, W nl DFS Ã! W km e [k], m integer. DFS ex [n] Ã! e [k l], l integer Duality: Since e [k] ex [n] W kn 1 ex [n] W kn 1 ( ex [n]) W k( n) 1 ( ex [ m]) W km m0 1 ( ex [ m]) W km m0 5 µ W n W n m n
6 and Thus, if then ex [n] 1 1 k0 e [k] W kn. ex [n] DFS Ã! e [k] e [n] DFS Ã! ex [ k]. ote: This property is unique for DFS! Symmetry Properties: See items 9-17 in Table Periodic Convolution: 1 m0 Table 8.1 ex 1 [m] ex 2 [n m] DFS Ã! e 1 [k] e 2 [k] Pf: The Fourier series coe cients of P 1 m0 ex 1 [m] ex 2 [n m] are à 1 1! ex 1 [m] ex 2 [n m] otes: W kn 1 ex 1 [m] m0 e 2 [k] e 1 [k]. m0 à 1! ex 2 [n m] W k(n m) {z } 2 [k] (since x 2 [n] is periodic with period ) 1. We shall denote the periodic convolution ex 1 [n] ~ P ex 2 [n] 1 m0 ex 1 [m] ex 2 [n m]. W km 2. ex 1 [n] ~ P ex 2 [n] ex 2 [n] ~ P ex 1 [n], i.e., ~ P is commutative. 3. The periodic convolution of two periodic sequences of period is also a periodic sequence of period. 6
7 4. Procedure comparison between periodic convolution and aperiodic convolution: Aperiodic: (Convolution Sum) Periodic: Ex: Periodic Convolution Fig. 84-B1 Fig. 85-F1 Fig The DT Fourier Transform of Periodic Signals ² Recall that absolute summability is a su cient condition for the existence of DT Fourier transform. o periodic sequence is absolutely summable! ² Defn: The DT Fourier transform e (e jω ) of a periodic sequence ex [n] is de ned as an impulse train in the frequency domain with impulse values proportional to the DFS coe cients of ex [n]. Mathematically, e e jω, where δ (ω) is an impulse. 1 k 1 µ 2π e [k] δ ω 2πk ow, denoting x as a value smaller than but extremely close to x, Z 2π 1 e e jω e jωn dω 2π 0 1 Z 2π 1 µ 2π 2π 0 e [k] δ ω 2πk e jωn dω k Z 2π µ e [k] δ ω 2πk e jωn dω k 1 0 Ã Z 2π µ ω 2πk ½ e jωn e j 2π dω nk, if 0 2π k < 2π 1 0 δ 1 k0 ex [n]. e [k] e j 2π nk 7!
8 Thus, in a sense that and ex [n] ex [n] 1 2π e e jω e [k] DT FT Ã! e e jω Z 2π 1 k e e jω e jωn dω µ 2π e [k] δ ω 2πk ex [n] e j 2π nk ~ ² ote: This e (e jω ) does not converge in any normal sense. However, this form ~ is useful and convenient for analysis purpose. In other words, the introduction of impulses permits us to include periodic sequences formally within the framework of discrete-time Fourier transform analysis. ² Ex: The DTFT of a Periodic Discrete-Time Impulse Train Consider the periodic discrete-time impulse train ow, its DFS coe cient is ep [n] and the corresponding DTFT is ep (e jω ) 1 r 1 δ[n r]. ep [k] 1 for all k 1 k 1 µ 2π δ ω 2πk. ² Consider a nite-length signal x[n] such that x[n] 0 except in the interval 0 n 1. ow, we produce a periodic signal ex[n] by (aperiodically) convolving x[n] with the periodic impulse train ep [n] as ex[n] x[n] ep [n] Fig r 1 x[n r]. ()
9 ote that we can express x[n] in terms of ex[n] as x [n] ½ ex [n], 0 n 1. From (), the DT Fourier transforms of x[n] and ex[n] are related by e e jω e jω ep e jω e jω 1 1 k 1 Thus, e [k] is related to ³ ³ e [k] k 1 2π ³ e j 2π k e j 2π k by 2π δ e j 2π k µ ω 2πk µ δ ω 2πk e jω ω 2π k Thus, e [k] is obtained by sampling the DTFT (e jω ) at equally spaced frequencies between ω 0 and ω 2π with a frequency spacing of 2π. This is exactly the same relation as we have shown in Section 8.1. ² Ex: Relationship Between the DFS Coe cients and the DTFT of One Period Consider the nite-length signal with 10 ½ 1, 0 n 4 x [n] 0, 5 n 9 ) e jω 4 e jωn e j2ω sin 5 ω 2 sin ω 2 ) e [k] e jω ω 2π k 10.. e sin π k j2π k/5 2 sin π k 10 where e [k] is the DFS coe cient of the periodic signal ex[n], given by ex[n] 1 x[n 10r]. r 1 Figs
10 8.4 Sampling The DT Fourier Transform ² Consider an aperiodic sequence x [n] with DT Fourier transform (e jω ) F fx [n]g, i.e., e jω 1 x [n] e jωn. n 1 ote that x [n] may be of nite duration or of in nite duration. Let us pick one number and sample (e jω ) at equally spaced frequencies between ω 0 and ω 2π with a frequency spacing of 2π as e [k] ³ e jω ω 2π e j 2π k (z)j ze j 2π k which is periodic with period. Fig. 8.7 ² ow, let ex [n] be the periodic sequence expanded by e [k], i.e., Since ) ex [n] 1 ex [n] k k0 1 m 1 1 k0 m 1 1 m 1 x [m] 1 e [k] W kn 3 x [m] e j 2π {z km 5 } W kn W km x [m] W k(m n) 1 k0 W k(m n). 1 1 W kα k0 ( 1, if α integer 1 1 W α 1 W α 1 k 1 δ [α k] 10
11 ) ex [n] ) ex [n] 1 m 1 x [n] 1 k 1 x [m] 1 k 1 1 k 1 x [n k] δ [(n m) k] δ [n k] i.e., ex [n] represents the periodic sequence resulting from repeating x [n] every samples. Figs ² otes: 1. In other words, the samples of DT Fourier transform of x [n] can be thought of as DFS coe cients of ex [n]. 2. If x [n] is of nite duration, it can be recovered from ex [n] provided that is larger than its duration. Speci cally, x [n] ½ ex [n], 0 n 1 0, elsewhere. 3. Given a nite-length x [n], pick larger than or equal to the length and conduct the transforms Fig. 87-B1 Under this circumstance, e [k] for k 0, 1,..., 1 is called the discrete Fourier transform (DFT) of the nite-length sequence x [n]. (a) Key Point: x [n] is one period of ex [n] (may not start from n 0!) since 1 ex [n] x [n k]. k 1 (b) ote: has to be chosen not smaller than the length of x [n]! 11
12 8.5 Fourier Representation of Finite-Duration Sequences: The DFT ² Consider a nite-duration sequence x [n] with x [n] 0 outside the range 0 n 1. De ne ex [n] 1 r 1 x [n r] which is periodic with period. Then, it is clear that x [n] can be retrieved from ex [n] x [n] ½ ex [n], 0 n 1. ² An alternative form to de ne ex [n] is ex [n] x [(n modulo )], x [((n)) ]. A cylindrical visualization of ex [n] x [((n)) ] is Fig. 88-B1 ² Let e [k] be the DFS of ex [n] and de ne [k] ) e [k] [((k)) ]. ½ e [k], 0 k 1. Since e [k] 1 ex [n] 1 1 ex [n] W kn 1 k0 e [k] W kn x [n] W kn 1 1 k0 [k] W kn ) [k] ½ P 1 x [n] W kn, 0 k 1 P 1 k0 ½ 1 kn [k] W x [n], 0 n 1 12 ( ). (+)
13 ( ) and (+) form a Discrete Fourier Transform pair; ( ) is called an analysis equation, while (+) is called a synthesis equation. We shall use the nomenclature x [n] Ã! DFT [k] and [k] DFT fx [n]g throughout. ² Usually, we use for notational brevity and [k] x [n] k0 x [n] W kn [k] W kn as the de nition of the DFT pair, keeping in mind that [k] 0 for k /2 [0, 1] and x [n] 0 for n /2 [0, 1]. ² Ex: The DFT of a Rectangular Pulse Consider the nite-duration sequence with length ( 5) ½ 1, 0 n 4 x [n] 0, 5 n 1 Fig for Properties of DFT Fig for 10 ² Consider the properties of the DFT for nite-duration sequences. We let x i [n] Ã! DFT i [k] de ned over length i throughout Linearity: If then x 3 [n] ax 1 [n] + bx 2 [n] 3 [k] a 1 [k] + b 2 [k] where x 1 [n] and x 2 [n] are of length 1 and 2 respectively; and x 3 [n] is of length 3 max f 1, 2 g. The DFT s 1 [k], 2 [k], and 3 [k] are de ned over length 3. Fig. 89-B1 13
14 In other words, the shorter sequence should be zero-padded, and the corresponding DFT de ned over the largest length Circular Shift of A Sequence: Let and Then, ex [n] x [((n)) ] ex 1 [n] ex [n m]. x 1 [n] ½ ex1 [n] ex [n m], 0 n 1 8 < : 8 < : ex [n m], m n 1 ex [n m + ], 0 n m 1 x [n m], m n 1 x [n m + ], 0 n m 1 (ex [n] x [((n)) ]) Thus, we say Fig x 1 [n] ½ x [((n m)) ], 0 n 1 a circular shift of x [n] by m samples. Since [k] 1 1 x [n] W kn 14
15 In summary, ) 1 [k] W km W km W km W km x 1 [n] W kn for 0 k 1 x [((n m)) ] W kn ex [n m] W kn W km [k]. 1 m n 0 m periodic periodic {z } n 0 n m 1 n ex ex hn 0i W kn0 {z } {z } hn 0i W kn0 x [((n)) ] W kn x [n] W kn x 1 [n] x [((n m)) ], 0 n 1 DFT Ã! W km [k] Duality: Since x [n] 1 1 k0 [k] W kn 15
16 we have Thus, if then [k] x [n] W kn 1 n n 0 0 (ex [n]) W kn 0 1 h ex n 0i ( C A W k n 0) {z } {z } periodic periodic ³ ex h 0i ( n n W k 0) 1 n n 0 0 ³ h³³ i x n 0 W kn0 ³ h³³ i x n 0 W kn0. x [n] DFT Ã! [k] [n] DFT Ã! x [(( k)) ] for 0 k 1. Fig ³ n 0 n Symmetry Property: Since ex [n] x [((n)) ] e [k] [((k)) ] and we have ex [n] DFS Ã! e [ k] ex [ n] DFS Ã! e [k] x [((n)) ] for 0 n 1 DFT Ã! [(( k)) ] for 0 k 1 ) x [n] for 0 n 1 DFT Ã! [(( k)) ] for 0 k 1 16
17 and x [(( n)) ] for 0 n 1 DFT Ã! [((k)) ] for 0 k 1 ) x [(( n)) ] for 0 n 1 DFT Ã! [k]for 0 k 1 See Table 8.2 for more symmetry properties Circular Convolution: Let ex 3 [n] ex 1 [n] ~ P ex 2 [n]. Then Table 8.2 x 3 [n], ex 3 [n] for 0 n 1 ex 1 [n] ~ P ex 2 [n] 1 m0 ex 1 [m] ex 2 [n m] 1 x 1 [((m)) ] x 2 [((n m)) ] m0 1 x 1 [m] x 2 [((n m)) ] for 0 n 1. m0 This last expression is called the circular convolution between x 1 [n] and x 2 [n] (both nite-duration sequences of length ). We shall denote Furthermore, x 3 [n] 1 x 3 [n] x 1 [n] ~ x 2 [n], x 1 [m] x 2 [((n m)) ]. 1 m0 m0 ex 1 [n m] ex 2 [m] for 0 n 1 1 x 1 [((n m)) ] x 2 [m] m0 which tells us x 3 [n] x 1 [n] ~ x 2 [n] x 2 [n] ~ x 1 [n]. i.e., ~ is commutative µ since ex1 [n] ~ P ex 2 [n] ex 2 [n] ~ P ex 1 [n] 17
18 ² Ex: Circular Convolution With a Delayed Impulse Sequence Let x 2 [n] be a nite-duration sequence of length and x 1 [n] δ [n n 0 ] for 0 < n 0 < ) 1 [k] W kn0. De ning 3 [k] W kn0 2 [k], we nd from the circular shifting property, x 3 [n] x 2 [((n n 0 )) ] for 0 n 1. Also, x 1 [n] ~ x 2 [n] Thus, this example implies 1 m0 x 1 [m] x 2 [((n m)) ] 1 δ [m n 0 ] x 2 [((n m)) ] m0 x 2 [((n n 0 )) ] for 0 n 1. x 1 [n] ~ x 2 [n] DFT Ã! 1 [k] 2 [k]. ² In general, letting max f 1, 2 g, we have for x i [n] DFT Ã! i [k] de ned over length that 1 [k] 2 [k] 1 n n 10 n 20 1 n x 1 [n 1 ] W kn n 1 n n 2 0 x 2 [n 2 ] W kn 2 ex 1 [n 1 ] ex 2 [n 2 ] W k(n 1+n 2 ) ex 1 [n 1 ] nn q 1 ex 2 [n n 1 ] q x 1 [n 1 ] x 2[((n n 1 )) ] 1 x 1 [n 1 ] x 2 [((n n 1 )) ] {z } Ã 1 n 10 periodic W kn x 1 [n 1 ] x 2 [((n n 1 )) ] x1 [n] ~ x 2 [n] W kn. µ W kn {z} periodic! n n 1 + n 2 ) n 2 n n 1 W kn 18
19 Similarly, one can show that Thus, x 1 [n] ~ x 2 [n] 1 1 k0 1 [k] 2 [k] W kn. x 1 [n]~ x 2 [n] for 0 n 1 DFT Ã! 1 [k] 2 [k] for 0 k 1. ² A duality to the previous property is Pf: x 1 [n] x 2 [n] 1 x 1 [n] x 2 [n] DFT Ã! 1 1 [k] ~ 2 [k] µ 1 µ 1 µ µ 1 µ 1 1 k [k 1 ] W k 1n k 10 k k k 1 kk 1 k k 1 + k 2 ) k 2 k k k 1 0 k0 2 1 k0 1 k0 Ã 1 k k 20 2 [k 2 ] W k 2n e 1 [k 1 ] e 2 [k 2 ] W (k 1+k 2 )n periodic e 1 [k 1 ] q 1 [k 1 ] periodic e 2 [k k 1 ] q 2 [((k k 1 )) ] W kn 1 [k 1 ] 2 [((k k 1 )) ] W kn 1 [k 1 ] 2 [((k k 1 )) ] 1 [k] ~ 2 [k] W kn 1 IDFT 1 [k] ~ 2 [k] ª. Similarly, the other direction 1 1 [k] ~ 2 [k] DFT fx 1 [n] x 2 [n]g can be proved. QED! W kn ² See Table 8.2 for a complete list of DFT properties (prove them for your own fun). 19
20 8.7 Computing Linear Convolution Using The DFT ² Since e cient algorithms exist for computing DFT (e.g., Fast Fourier Transform (FFT)), it is desirable to compute linear convolution of two sequences (e.g., implementing an LTI system) by DFT approach. That is, we want to perform Fig. 94-B1 Question: How is x 1 [n] x 2 [n] (linear convolution) related to x 1 [n] ~ x 2 [n] (circular convolution)? ² ow, consider causal sequences x 1 [n] of length L and x 2 [n] of length P. We want to nd the output of the linear convolution x 3 [n] x 1 [n] x 2 [n]. Taking discrete-time Fourier transform both-sided, Since for max fl, P g, 3 e jω 1 e jω 2 e jω. [k] ½ e [k], 0 k 1 ( ³e k j 2π, 0 k 1 ) 3 [k] 3 ³ e j 2π k 1 ³ e j 2π k 2 ³ e j 2π k ) 3 [k] 1 [k] 2 [k] for 0 k 1. Denoting we have x 3p [n] DFT Ã! 3 [k] x 3p [n] x 1 [n] ~ x 2 [n] for 0 n 1 where max fl, P g and x 1 [n] DFT Ã! DFT 1 [k], x 2 [n] Ã! 2 [k]. points points for 0 k 1 ² Question: How large should be, such that x 3 [n] x 3p [n] for all n? Answer: 20
21 1. x 3 [n] x 1 [n] x 2 [n] minfl 1,ng mmaxf0,n P +1g ) x 3 [n] is nontrivial when 1 m 1 x 1 [m] x 2 [n m] x 1 [m] x 2 [n m] 0 min fl 1, ng max f0, n P + 0 m L 1 0 n m P 1 n P + 1 m n or equivalently when 0 n L + P (a) Assume L P without loss of generality: B If n L 1, then L 1 n P + 1 ), n L + P 2. If L 1 > n P 1, then n n P + 1 ) P 1 and n 0. A (b) The same result can be obtained for P L similarly. Thus, x 3 [n] has length L + P 1 and is nontrivial when n 0, 1,..., L + P 2. ) has to be no smaller than L + P 1!! 2. ext, for 0 n 1, Fig x 3p [n] x 1 [n] x 2 [((n m)) ]. m0 If we choose L + P 1, then (a) x 3p [n] L+P 2 m0 x 1 [m] x 2 ((n m))l+p 1 L 1 x 1 [m] x 2 ((n m))l+p 1. m0 ((n m)) L+P 1 n ½ m mod L + P 1 n m, 0 m n n m + (L + P 1), n < m n + (L + P 1) noting that 0 n L + P 2 and 0 m L A
22 (b) x 2 [n m] is nontrivial for n m 2 [0, P 1] ) m 2 [n P + 1, n]. (c) x 2 [n m + (L + P 1)] is nontrivial for 0 n m+(l + P 1) P 1. ) n + L m n + L + P 1. This case is not possible since 0 n L + P 2 and 0 m L 1. (d) Therefore, x 3p [n] minfl 1,ng x 1 [m] x 2 [n m] mmaxf0,n P +1g x 3 [n] for 0 n Trivially, x 3p [n] x 3 [n] 0 for n /2 [0, 1]. In summary, if we let L + P 1, then the linear convolution of two nite length sequences x 1 [n] and x 2 [n] of length L and P, respectively, is equal to their circular convolution with. In this case, one can use DFT and inverse DFT to obtain the linear convolution x 1 [n] x 2 [n]. otes: 1. Let L + P 1. Then we can perform Fig. 97-F1 2. An Illustration: Since x 3p [n] P 1 r 1 x 3 [n r] for a predetermined period Fig Fig ² Implementing LTI Systems Using The DFT Consider a causal FIR system with h [n] of length P and an semiin nite-length causal sequence x [n]. Let x [n] be segmented into nonoverlapping blocks of length L, i.e., x [n] 1 x r [n rl] 1 rl n<(r+1)l r0 where 1 condition 1 if the condition is met and 1 condition 0 otherwise. Fig. 97-F2 22
23 ow, we want to implement y [n] x [n] h [n] 1 x r [n rl] 1 rl n<(r+1)l h [n] r0 " 1 1 r0 m L 1 4 r0 m L 1 4 r0 m 0 0 x r [m rl] 1 rl m<(r+1)l h [n m] x r hm 0i h h n m 0 rli 3 5 ³ m 0 m rl x r hm 0i h h (n rl) m 0i3 5 1 x r [n] h [n rl]. r0 If we choose L + P 1, we have # y [n] 1 x r [n] ~ h [n rl]. r0 ote here that x r [n]~ h [n rl] is nontrivial for 0 n rl 1. ote that y [n] x [n] h [n] 1 y r [n rl] with y r [n rl] x r [n] ~ h [n rl]. This implies an implementation approach called the overlap-add approach. The following is an illustration: r0 Fig Fig An alternative approach is overlap-save approach. Assuming P < L, the approach is to divide x [n] into overlapping blocks of length L with each block overlapping the preceding block by P 1 symbols, i.e., the rth block containing x r [n] x [n + r(l P + 1) P + 1] for 0 n L 1. 23
24 The L-point circular convolution of x r [n] with h[n] is denoted by y rp [n]. ow, deleting the rst P 1 symbols in y rp [n] yields ½ yrp [n], P 1 n L 1 y r [n]. Thus, the output sequence y[n] x[n] h[n] is given by y [n] 1 y r [n r(l P + 1) + P 1]. i0 (Prove it yourself) The following is an illustration: Fig ² ote: E cient algorithms, like Fast Fourier Transform (FFT), can be applied to implement the circular convolution, and, thus, the LTI system operation! 24
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