EE 477 Digital Signal Processing. 4 Sampling; Discrete-Time

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1 EE 477 Digital Signal Proceing 4 Sampling; Dicrete-Time

2 Sampling a Continuou Signal Obtain a equence of ignal ample uing a periodic intantaneou ampler: x [ n] = x( nt ) Often plot dicrete ignal a dot or lollypop : Time index, n EE 477 DSP Spring 2007 Maher 2

3 Sampling a Sinuoid Dicrete time inuoid via ampling: x[ n] ( ω nt + φ ) = Aco( ωˆ + φ ) = x( nt ) = Aco n Note that A, co, φ are the ame. Dicrete-time radian frequency: ω ˆ= ωt Note that T cannot be deduce from x[n] alone! EE 477 DSP Spring 2007 Maher 3

4 Recontruction?? It i poible to recontruct a continuou-time ignal from it dicretetime ample, but with retriction. The ampling theorem tate that a ignal can theoretically be recontructed from it ample a long a 1 f = 2 fmax T EE 477 DSP Spring 2007 Maher 4

5 Sampling Rate In hort, we mut ample at a rate at leat double the highet frequency component preent in the continuoutime ignal. Thi minimum ampling rate i called the Nyquit rate. Reult: continuou-time ignal mut be bandlimited prior to ampling in order to allow perfect recontruction. EE 477 DSP Spring 2007 Maher 5

6 Aliaing What happen if we don t obey Nyquit? Conider two ignal: x( t) y( t) = = Aco Aco ( 2πf t + φ ) ( 2π ( f + f ) t + φ ) (ame amplitude and phae, different freq) 0 0 EE 477 DSP Spring 2007 Maher 6

7 Aliaing (cont.) Now ample with period T : x y [ n] = Aco( 2πf ) 0nT + φ [] n = Aco( 2π ( f + f ) nt + φ ) = = Aco 2πf Aco ( 2πf nt + φ ) = x( t) nt + 2πf nt πn + φ EE 477 DSP Spring 2007 Maher 7

8 Aliaing (cont.) Note that the ame ampled equence occur for both x(t) and y(t) even though they have different frequencie: one ignal i an alia of the other. Further, note that infinite number of aliae ince ame dicrete-time equence for: f = f0 ± kf, k = 0,1,2,... EE 477 DSP Spring 2007 Maher 8

9 Folding Alo can find aliae correponding to the negative frequency component: w( t) = Aco 2π f + kf nt φ = = ( ( ) ) Aco 2πf Aco + 2πf nt ( 2πf nt + φ ) = x( t) nt 2πn φ EE 477 DSP Spring 2007 Maher 9

10 Spectral View of Sampling The effect of ampling i to create image of the continuou-time pectrum centered at multiple of the ampling frequency: -2f -f 0 f 2f 0 f o -f -f f f -2f -2f 2f 2f EE 477 DSP Spring 2007 Maher 10

11 Spectral View (cont.) We can recontruct the continuou ignal by removing (filtering) the image and keeping the baeband image: -2f -f 0 f 2f -f -f f f -2f -2f 2f 2f EE 477 DSP Spring 2007 Maher 11

12 Aliaing What if f 0 > f /2? Sampling till create image, but now the baeband image i not the expected original ignal, but actually aliae. -2f -f 0 f 2f -2f -3f -f -2f -f 0 f f 0 2f f 3f 2f EE 477 DSP Spring 2007 Maher 12

13 Recontruction==Interpolation The recontruction proce can be thought of a interpolating between the dicrete-time ample. Variou interpolation approximation can be conidered: hold lat value, connect the dot (linear), fit a mooth polynomial curve, etc. Optimal recontruction require a proce that retain only the baeband: a perfect lowpa filter. EE 477 DSP Spring 2007 Maher 13

14 Concept: Pule-overlap Interpolation Conider contructing the continuou waveform by hifting and caling a et of pule one centered per dicretetime ample then um them all up. -T /2 T /2 Time index, n EE 477 DSP Spring 2007 Maher 14

15 Pule Overlap (cont.) Triangular pule = linear interpolation Similar for higher-order interpolation -T T Time index, n EE 477 DSP Spring 2007 Maher 15

16 Recontruction via Filtering The pule overlap cheme implement time domain convolution. Time domain convolution i equivalent to frequency domain multiplication We want a perfect rectangle (low pa) in the frequency domain: thi correpond to a inc pule in time domain: π -2T -T T 2T p ideal () t in T π t T EE 477 DSP Spring 2007 Maher 16 = t

17 Overampling Interpolation i eaier of ample are cloe together: T i very mall Small T mean very high f From a pectral viewpoint, thi overampling mean that f max << f Filter need not be perfect -2f -2f -f -f 0f 0 f f 2f 2f EE 477 DSP Spring 2007 Maher 17