Roadmap for Discrete-Time Signal Processing

Transcription

1 EE 4G Note: Chapter 8 Continuou-time Signal co(πf Roadmap for Dicrete-ime Signal Proceing Dicrete-time Signal (Section 8.) Sampling Period econd (or ampling frequency / Hz) Denoted a n) or imply n) for n,,, If i too big, we have no idea of the original continuou-time ignal! Page 5-

2 EE 4G Note: Chapter 8 So increae the ampling frequency, but by how much? Surpriing reult by Nyquit (Section 8.) A continuou-time ignal, having no frequency component above f h Hz, can be completely recovered by ample taken at a ampling rate greater than or equal to f h Hz. hi i called the Nyquit rate. Recontruction formula: π ( t n ) in( ) n ) n π ( t n ) n t n n )inc Quantization (Section 8.) Not only time can be dicretized, ignal level can too (and mut to be tored in computer). x ( n ) A 3-bit computer can tore 3 4,94,967, 96 level. I it good enough? Similaritie between continuou-time and dicrete-time analyi PROPERIES CONINUOUS-IME DISCREE-IME Sytem Propertie Sytem Equation Complex Frequency ranform Linearity, Cauality, Stability Differential Equation d y( + 3y( dt Laplace ranform X t ( ) e dt Linearity, Cauality, Stability (8.4) Difference Equation (8.4) y [ n ] y[( n ) ] x[ n ] Z-tranform (8.3) X ( z) n n ) z n Page 5-

3 EE 4G Note: Chapter 8 ranfer function Fourier ranform Fourier Serie H ( ) Continuou-ime Fourier ranform (CF) jωt t e dt X ( jω ) ) Continuou-ime Fourier Serie for Periodic Signal X k e π jk t dt ( ) z H z z + 4z + Dicrete-time Fourier ranform (DF) (8.4) X ( jω ) n n ) e jωn Dicrete Fourier ranform (DF) for Finite-Duration Signal (4-3,-,-4) X k N n n) e πk j n N Synthei echnique Filter Deign: Again Low-pa, High-pa, Band-pa, Band-reject Baic trategy : tart with analog prototype and then map to dicrete-domain.. Infinite Impule Repone (IIR) Filter Deign (Section 9-4) - impule repone goe on forever : h( n ), n,,,.... Finite Impule Repone (FIR) Filter Deign (Section 9-5) finite impule repone : h ( n ) [ 5 ] Computational echnique Fat Fourier ranform (Section -4) i a fat algorithm to compute DF. he baic idea i to the recurive nature of the dicrete-time complex exponential to reduce the complexity of the algorithm. n Page 5-3

4 EE 4G Note: Chapter 8 Part one: A/D and D/A Chapter 8 Dicrete-ime Signal and Sytem 8.A Analog-to-Digital Converion Sampling Operation Sampling Model: n x ( δ ( t n ) Here, our ampling function p ( i a periodic ignal compoed of a train of delta function. n p ( t ) δ ( t n ) Why x (, intead of jut uing the value of x ( at,,,?. Almot the ame x ( n ) δ ( t n ) t n t n Page 5-4

5 EE 4G Note: Chapter 8. he impule retain energy of the ample behave like a real ignal! x ( dt n ) δ ( t n ) dt n n ) For analyi purpoe, we ue x ( to repreent the dicrete-time ignal x (n ) ( x ( i the continuou-time urrogate of x (n ) ). Spectrum of X and it relationhip to X Multiplication in ime Convolution in Frequency x ( p( X X [ j( ω υ)] P( jυ) dυ From able 4- (Ch.4), mπ P( jω) F[ δ ( t n )] δ ω n m π - another impule train (in frequency domain) with period ω πf, our ampling frequency hu, X ( jω) X [ j( ω υ)] δ ( υ mω ) m dυ X ( jω jmω ) m - Compare thi with the impule train. X become a periodic pectrum with period ω. Pictorially, if X look like X(jω) -ω h ω h What doe X look like? Page 5-5

6 EE 4G Note: Chapter 8 wo cae: ω Cae : ωh : ucceive copie of X do not overlap X (jω) -ω -ω h -ω -ω +ω h -ω h ω h ω -ω h ω ω +ω h ω It i trivial to ee that we can recover X by multiplying X with the ideal low-pa filter with bandwidth ω /. H(jω) -ω / ω / ω Cae : ω h > : ucceive copie of X overlap X (jω) -ω -ω -ω h ω h ω ω No way to recontruct X. Nyquit ampling theorem: A ignal, having no frequency component above ω h rad/, i completely pecified by ample that are taken at a uniform ampling rate greater than or equal to ω h rad/. hi i called the Nyquit Sampling Rate. Page 5-6

7 EE 4G Note: Chapter 8 he phenomenon of ampling under the Nyquit rate i called ALIASING. In time domain: At the time of ampling, indicated by vertical line, the two ignal hown both agree with the obervation. Either of thee ignal could produce the indicated obervation, hown by circled point, o the two ignal are aid to be "aliaed." In frequency domain: Sine wave of f Hz X(jω) x ( ampling at f <f -πf X (jω) πf ω -4πf -πf -πf πf πf 4πf If our ear can be model a a low-pa filter with bandwidth around f Hz, EAR(jω) ω ω thi i what we will hear: the original plu unwanted lower-frequency component. EAR(jω)X (jω) -πf πf ω Page 5-7

8 EE 4G Note: Chapter 8 Original Image: Downample by a factor of two: look at the low frequency pattern around the tripe. If ampling frequency i fixed (for example CD), the original ignal i low-paed firt to enure that no aliaing occur. hi proceing i called anti-aliaing. Page 5-8