5.1 The Discrete Time Fourier Transform

Transcription

1 The Discrete Time ourier Transform ourier (or frequency domain) analysis the last Complete the introduction and the development of the methods of ourier analysis Learn frequency-domain methods for discrete-time signals and systems Outline 5. The Discrete Time ourier Transform 5.2 The ourier Transform for Periodic Signals 5. The Discrete Time ourier Transform Development of the ourier Transform Representation Analogous to continuous time case Aperiodic signal Construct periodic signal with over one period Period over any finite time interval ourier series representation of converges to ourier transform representation of General sequence with outside n Properties of the Discrete Time ourier Transform 5.4 The Convolution Property 5.5 The Multiplication Property 5.6 Duality 5.7 requency Response and Linear Constant Coefficient Difference Equations 2 Corresponding periodic signal n 2 n Obviously: for

2 34 35 requency domain: a k e jk(/)n S a k or k<> a k n<> 2 e jk(/)n n e jn k X(e j ) Sample spacing: / Therefore k<> ow: ( ) X(ejk )e jk(/)n e jk(/)n e jk(/)n k<> k Summation integration k, d Summation over intervals of width / integration interval X(e j )e jn periodic with period any interval of length X(e jk )e jk n Discrete-time ourier transform X(e j ) e jn and inverse discrete-time ourier transform X(e j )e jn d X(e j ): referred to as the ourier transform or the spectrum of Short-hand notation X(e j ) X(e j ) {} {X(e j )} ote: ourier series coefficients a k of periodic signal are equally spaced samples of ourier transform X(e j ) of aperiodic signal, where over one period and zero otherwise. Differences to continuous time ourier transform Periodicity of discrete time ourier transform X(e j ) inite interval of integration in inverse ourier transform Slowly varying signals: nonzero spectrum around k, k Z ast varying signals: nonzero spectrum around + k, Z

3 Signal a n u[n], a < ourier transform X(e j ) e jn ae j Discrete-time ourier transform pair a n u[n] a n u[n]e jn ae j Magnitude of spectrum X(e j ) ae j + a2 2acos a : u[n] very slowly varying spectrum concentrated around k, k Z X(e j ) /( a) (ae j ) n n a : ( ) n u[n] very fast varying spectrum concentrated around + k, k Z 2. Unit impulse ourier transform X(e j ) δ[n] /( + a) δ[n] δ[n] X(e j ) /( a) δ[n]e jn {δ[n]} /( + a) n

4 Shifted unit impulse 4. Rectangular pulse δ[n n ] ourier transform X(e j ) δ[n n ]e jn e jn {, n, n > δ[n n ] δ[n n ] e jn X(e j ) n ourier transform 2 X(e j ) e jn e j n n e jn e j e j(2 +) e j ej e j( +) e j/2 (e j/2 e j/2 ) sin( + /2) sin(/2) X(e j ) {, n, n > sin(( + /2)) sin(/2) n n n n X(e j )

5 4 4 Convergence of discrete time ourier transform ourier transform also valid for signals with infinite duration if the associated infinite sum converges. Sufficient conditions for convergence is absolutely summable < or has finite energy 2 < o convergence issues associated with inverse discrete-time ourier transform o Gibbs phenomenon 5.2 The ourier Transform for Periodic Signals Periodic discrete-time signals do not meet the Convergence Conditions of Section 5., but also have discrete-time ourier transforms, which can directly be constructed from their ourier series. Consider with ourier transform X(e j ) δ( l) X(e j ): Impulse train l X(e j ) Inverse ourier transform l e j n δ( l)e jn d e j n l δ( l)

6 42 43 Accordingly: periodic signal with period a k e jk(/)n S a k k<> X(e j ) X(e j ) k<> k a k l δ( k/ l) a k δ( k/) ourier transform of a periodic signal train of impulses occurring at multiples of the fundamental frequency / area of impulse at k/ is times the kth ourier series coefficient k : a a a a a. Sine function sin( n) ( e j n e ) j n 2j k : k : ( + ) ( ) a a + a ( + ) ( ) a a + Periodic if m/ ourier transform X(e j ) j δ( l) + j δ( + l) l l X(e j ) j : a a a a a a a + j a

7 Cosine function cos n 2 ej n + 2 e j n Periodic if m/ ourier transform X(e j ) δ( l) + δ( + l) l l X(e j ) 2 X(e j ) 2 n Impulse train sequence δ[n k] ourier series coefficients a k n<> n<> l k e jk(/)n δ[n l]e jk(/)n ourier transform X(e j ) k δ( k/)

8 Properties of the Discrete Time ourier Transform Periodicity Discrete time ourier transform is always periodic in with period X(e j(+) ) X(e j ) Different from continuous time ourier transform! Linearity ourier transform {a + by[n]} a{} + b{y[n]} inverse ourier transform {cx(e j ) + dy (e j )} c {X(e j )} + d {Y (e j )} Time Shifting If then Proof: {x[n n ]} x[n n ] e jn X(e j ) e jn X(e j ) x[n n ]e jn n n x[n ]e j(n +n ) x[n ]e jn e jn X(e j ) requency Shifting If then e j n X(e j ) X(e j( ) ) Proof: {X(e j( ) )} X(e j( ) )e jn d X(e j )e j( + )n d e j n X(e j )e j n d e j n Periodicity property + frequency shifting property special relationship between ideal lowpass and ideal highpass discrete time filters

9 48 49 Time Reversal Lowpass filter with cutoff frequency c requency response H lp (e j ) c H lp (e j ) Highpass filter with cutoff frequency c requency response H hp (e j ) c H lp (e j ) If then Proof: Y (e j ) m n x[ n] m X(e j ) X(e j ) X(e j ) y[n]e jn x[m]e jm m x[ n]e jn x[m]e j( )m + c c requency domain relation H hp (e j ) H lp (e j( ) ) Impulse response inverse ourier transform of frequency response of an LTI system h hp [n] e jn h lp [n] ( ) n h lp [n] Conjugation and Conjugate Symmetry If then x [n] Real : conjugate symmetry X(e j ) X (e j ) X(e j ) X (e j )

10 5 5 Conjugation + time reversal properties irst order differencing Re{Ev{}} + Re{Od{}} + jim{ev{}} + jim{od{}} X(e j )Re{Ev{X(e j )}} + Re{Od{X(e j )}} + jim{ev{x(e j )}} + jim{od{x(e j )}} If then x[n ] X(e j ) ( e j )X(e j ) Consider Observe where With a n, a < z[n] + z[ n] δ[n] z[n] and time reversal property inally z[ n] z[n] a n u[n] ae j ae j X(e j ) ae + j ae j a 2 2acos() + a 2 real and even X(e j ) real and even ollows immediately from linearity and time-shifting properties Accumulation If then n m x[m] X(e j ) e j X(ej ) + X(e j ) Impulse train X(e j ): dc component Zero-mean (X(e j ) ): (see differencing property) X(e j ) e j k δ( k)

11 52 53 Unit step u[n] Known and Accumulation property X(e j ) u[n] δ[n] G(e j ) u[n] n m δ[m] e jg(ej ) + G(e j ) e + j k e + j k δ( k) k δ( k) δ( k) Decimation and Time Expansion Similar property(ies) to time and frequency scaling in continuous time case Consider X(e j ) Decimation: x[an], a I Proof: a X(e j(/a k/a) ) a k x[an] a X(e j(/a k/a) ) a n n/a k a a k e jn/a a n e j(/a k/a)n x[an ]e jn a e jkn/a k }{{}, if n/a Z, else {x[an ]}

12 54 55 Time Expansion (a I) Proof: { x[n/a], if n is a multiple of a x (a) [n], otherwise X (a) (e j ) x (a) [n] r X(e ja ) jn rn/a x (a) [n]e r x[r]e j(a)r X(e ja ) x (a) [ra]e jra and ourier transform {x[2n]} ( sin(5/4) 2 sin(/4) Decimation by 4 and ourier transform x[4n] δ[n] {x[4n]} ) sin(5( )/4) + sin(( )/4) {}. Decimation Rectangular pulse of length 2 5 x[2n] {x[2n]} and ourier transform Decimation by 2 {, n 2, n > 2 X(e j ) sin(5/2) sin(/2) x[2n] {, n, n > x[4n] {x[4n]}

13 Expansion Rectangular pulse of length 2 5 {, n 2, n > 2 and ourier transform X(e j ) sin(5/2) sin(/2) Expansion by 2 {, n, 2, 4 x (2) [n], else and ourier transform X(e j ) sin(5) sin() Expansion by 3 {, n, 3, 6 x (3) [n], else and ourier transform X(e j ) sin(5/2) sin(3/2) x (2) [n] x (3) [n] 3. ourier transform of sequence {, n 2k 2, n 2k +, k < 5 Observe y (2) [n] + 2y (2) [n ] with shifted rectangular sequence {, n < 5 y[n], else {} {x (2) [n]} {x (3) [n]}

14 58 59 Time-shifting and time-expansion properties y (2) [n] e j4sin(5) sin() Linearity and time-shifting properties inally 2y (2) [n ] Differentiation in requency j5 sin(5) 2e sin() X(e j ) e j4 ( + 2e j ) sin(5) sin(). ourier transform of Already known na n u[n], a < z[n] a n u[n] ae j Z(ej ) Using differentiation-in-frequency property nz[n] j dz(ej ) j d ( ) d d ae j we find na n u[n] ae j ( ae j ) 2 If then Proof: dx(e j ) d n X(e j ) j dx(ej ) d ( d ) e jn d jne jn 2. ourier transform of Since (n + )a n u[n], a < na n u[n] + a n u[n] we find from linearity property (n + )a n u[n] ( ae j ) 2

15 6 6 Parseval s Relation If then Proof: 2 X(e j ) 2 X(e j ) 2 d X (e j )e jn d X (e j ) e jn d X(e j ) 2 d In analogy with continuous time case X(e j ) 2 : Energy density spectrum of Given: ourier transform X(e j ) for of sequence Determine /2 3 2 X(e j ) /2 X(e j ) periodic? ourier transform has no impulses is aperiodic. real? Even magnitude and odd phase function, X(e j ) X (e j ) Symmetry is real. even? X(e j ) is not real valued Symmetry is not even. of finite energy? Integral of X(e j ) 2 over is finite Parseval has finite energy.

16 The Convolution Property Consider discrete-time LTI system with impulse response h[n] y[n] h[n] e jn is eigenfunction with eigenvalue H(e j ) {h[n]} (frequency response) e jn H(e j )e jn Guess: Proof: Y (e j ) H(e j )X(e j ) Convolution property y[n] h[n] Important to memorize Convolution in time domain domain Y (e j ) H(e j )X(e j ) multiplication in frequency requency response H(e j ) of discrete-time LTI system is discretetime ourier transform of impulse response h[n]. Impulse response completely characterizes LTI system. requency response completely characterizes LTI system. Y (e j ) { x[k]h[n k]} k k k k x[k] x[k] x[k]h[n k]e jn n x[k]e jk k n H(e j )X(e j ) h[n k]e jn h[n ]e j(n +k) h[n ]e jn Existence of frequency response convergence of discrete-time ourier transform LTI system is BIBO stable h[n] <, Discrete-time ourier transform converges Stable LTI systems have frequency response H(e j ) {h[n]}.

17 Discrete-time LTI system with impulse response requency response H(e j ) h[n] δ[n n ] δ[n n ]e jn e jn Input-output relation in frequency domain Y (e j ) e jn X(e j ) Time shifting property of ourier transform y[n] x[n n ] ote: time shift n in time domain unit magnitude and linear phase characteristic n in frequency domain 2. requency response of an ideal discrete time lowpass filter defined for {, H(e j c ), > c c H(e j ) c Impulse response h[n] Discrete-time ourier pair c H(e j )e jn d e jn d c ( c sinc c n ) H(e j ) {, c, > c H(e j ) periodic with period Observe: Impulse response h[n] is not causal. Impulse response h[n] has infinite duration. ideal lowpass not realizable As in the continuous-time case: Computing response of LTI system via frequency domain h[n] y[n] X(e j ) H(e j ) Y (e j ) Computational bottleneck : inverse ourier transform Solutions: sin( c n) n Look up table of (basic) ourier transform pairs (e.g., Table 5.2 in text book) Partial fraction expansion for ratio of polynomials

18 . Discrete-time LTI system with impulse response and input signal Output y[n]? h[n] α n u[n], α < β n u[n], β < Discrete-time ourier transforms of and h[n] X(e j ) βe j and H(e j ) αe j Discrete-time ourier transform of y[n] Y (e j ) H(e j )X(e j ) ( αe j )( βe j ) 66 (b) α β A ( αv)y (v) v/α B ( βv)y (v) v/β α α β β β α ( ) Y (e j α ) α β αe β j βe j α y[n] α β αn u[n] β α β βn u[n] ( α n+ β n+) u[n] α β We have Y (e j ) ( αe j ) 2 y[n] (n + )α n u[n] 67 Inverse discrete-time ourier transform (a) α β Write Y (e j A ) αe + B j βe j and consider generalized function (ourier pair derived in an example in Section 5.3, alternatively, write Y (e j ) j d ( ) α ej and use d αe j frequency-differentiation and time-shifting properties) Y (v) ( αv)( βv) A αv + B βv

19 Consider LTI system with ideal lowpass filters H lp (e j ) with cutoff frequency /4 (b) Lower path Convolution property ( ) n ( ) n W 4 (e j ) H lp (e j )X(e j ) w [n] H lp (e j ) w 2 [n] w 3 [n] (c) Combining top and lower path Linearity property y[n] Y (e j ) W 3 (e j )+W 4 (e j ) (H lp (e j( ) )+H lp (e j ))X(e j ) H lp (e j ) Relation between and y[n]? w 4 [n] requency response of overall system H(e j ) H lp (e j( ) ) + H lp (e j ) (a) Top path Since ( ) n e jn w [n] e jn from frequency shifting property Convolution property W (e j ) X(e j( ) ) W 2 (e j ) H lp (e j )X(e j( ) ) rom an example in Section 5.3: Ideal lowpass shifted by in frequency is ideal highpass filter Overall system passes both low and high frequencies and stops frequencies between these two passbands ilter with ideal bandstop characteristic, here stopband region /4 < < 3/4 H(e j ) Again frequency shifting property W 3 (e j ) W 2 (e j( ) ) H lp (e j( ) )X(e j( ) ) Periodicity of discrete time ourier transforms with period, W 3 (e j ) H lp (e j( ) )X(e j )

20 The Multiplication Property Analogous to continuous-time case: multiplication property Proof: y[n] x [n]x 2 [n] Y (e j ) X (e jθ )X 2 (e j( Θ) ) dθ Right hand side integral corresponds to periodic convolution can be evaluated over any interval of length. Y (e j ) y[n]e jn x [n]x 2 [n]e jn X (e jθ )e jθn dθ x 2 [n]e jn ( ) X (e jθ ) x 2 [n]e j( Θ)n dθ X (e jθ )X 2 (e j( Θ) ) dθ Spectrum of signal sin(n/2)sin(3n/4) 2 n 2 Write as product of two sinc functions ( n ) 2 sinc 3 ( ) 3n 2 4 sinc 4 rom multiplication property X(e j ) X (e jθ )X 2 (e j( Θ) ) dθ with periodic rectangular spectra defined in as {, /2 X (e j ), > /2 and X 2 (e j ) {, 3/4, > 3/4 Simplification of above periodic convolution by defining aperiodic lowpass ˆX (j) { X (e j ), <, otherwise

21 72 73 aperiodic convolution X(e j ) ˆX (jθ)x 2 (e j( Θ) ) dθ ˆX (jθ)x 2 (e j( Θ) ) dθ ˆX (j) X 2 (e j ) Result of convolution is ourier transform X(e j ) X(e j ) Duality o formal similarity between discrete-time ourier transform and its inverse X(e j ) e jn and X(e j )e jn d o duality property as in continuous-time case However: equations ormal similarity between discrete time ourier series k a k e jk(/)n and a k ormally define two periodic sequences g[n] S f[k] n e jk(/)n Changing roles of k and n in ourier coefficients series equation f[n] we find k g[k]e jk(/)n f[n] S g[ k] k g[ k]e jk(/)n

22 74 75 Duality of properties of discrete time ourier series Time and requency Shift and x[n n ] r e jm(/)n S a k e jk(/)n S a k m Convolution and Multiplication S x[r]y[n r] a k b k and y[n] S l a l b k l Duality often useful in reducing complexity of calculations involved in determining ourier series representations Periodic signal with a period of 9: sin(5n/9), n multiple of 9 9 sin(n/9), n multiple of Recall: ourier series coefficients b k of periodic square wave g[n] of length and a period (Chapter 3, page 84) {, n g[n], n > sin(k( + /2)/), k, ±, ±2,... sin(k/) b k 2 +, k, ±, ±2,... With 9 and 2 Duality 9 b k g[n] sin(5k/9) sin(k/9), k multiple of 9 5, k multiple of 9 9 S b k and b n S a k 9 g[ k] Sequence of ourier series coefficients with a period 9 { /9, k 2 a k, 2 < k 4

23 76 77 urthermore: ormal similarity between discrete time ourier transform X(e j )e jn d, X(e j ) and continuous time ourier series g(t) a k e jkt, a k T k T e jn x(t)e jk t dt ourier transform X(e j ) periodic with period ormally replace argument e j by t: X(t), periodic with period T, /T continuous-time ourier series X(t) e jtn k Observe: If x(t) periodic with period T and then a n g(t) S a k x[ k]e jk t X(e j ) g() Desired: discrete time ourier transform of sequence sin(n/2) ( n ) n 2 sinc 2 Continuous time signal g(t) with period T and ourier coefficients a k x[ k] (Chapter 3, page 74): {, t /2 g(t), /2 < t a k ( n ) 2 sinc a k x[k] 2 Duality ourier transform {, /2 X(e j ) g(), /2 < Verification: We have g(t) S a k sin(k/2) k or or sin(n/2) n /2 /2 ()e jn d X(e j ) g(t)e jkt dt /2 /2 /2 /2 ()e jn d ()e jkt dt

24 Summary of ourier series and transform expressions Continuous time Discrete time Time domain requency domain Time domain requency domain ourier Series ourier x(t) a k a k a k e jk t T x(t)e jk t a k e jk(/)n k T k e jk(/)n k continuous time discrete frequency discrete time discrete frequency periodic in time aperiodic in frequency periodic in time periodic in frequency x(t) X(j) X(e j ) X(j)e jt d x(t)e jt dt X(e j )e jn d e jn k Transform continuous time continuous frequency discrete time continuous frequency aperiodic in time aperiodic in frequency aperiodic in time periodic in frequency ote: continuous aperiodic discrete periodic requency Response and Linear Constant Coefficient Difference Equations Recall: Discrete-time system description by linear constant-coefficient difference equation aky[n k] k M bkx[n k] k requency response H(e j )? Apply e jn as input output y[n] H(e j )e jn Alternatively Convolution property H(e j ) Y (e j )/X(e j ) Apply ourier transform to difference equation + linearity and time-shifting properties { } { } M aky[n k] bkx[n k] k ak {y[n k]} k ake jk Y (e j ) k Consequently k M bk {x[n k]} k M bke jk X(e j ) k H(e j ) Y (ej ) X(e j ) M k bke jk k ake jk 79

25 8 8 Observe H(e j ) obtained from difference equation by inspection H(e j ) is a rational function in e j inverse ourier transform by partial-fraction expansion. Causal discrete-time LTI system characterized by requency response Impulse response y[n] ay[n ], a < H(e j ) ae j h[n] a n u[n] 2. Causal discrete-time LTI system characterized by y[n] 3 4 y[n ] + y[n 2] 2 8 requency response H(e j 2 ) 3 4 e j + 8 e j2 Partial-fraction expansion H(e j 2 ) ( 2 e j )( 4 e j ) e j 4 e j Impulse response h[n] 4 ( ) n u[n] 2 2 ( ) n u[n] 4 3. System of previous example and input ( ) n u[n] 4 Spectrum of output Y (e j ) H(e j )X(e j ) ( ) ( ) e j + 8 e j2 ( ) 4 e j 2 ( 2 e j )( 4 e j ) 2 Partial fraction expansion Y (e j ) B 4 e j + B 2 ( 4 e j ) 2 + B 2 2 e j with B 4, B 2 2, and B 2 8 Inverse ourier transform ( ( ) n y[n] 4 2(n + ) 4 ( ) n ( ) n ) u[n] 2