Part A:Sampling of Continuous-Time Signals Chapter 4 Part A ◆Introduction ●●● ●●● Sampling of Continuous-Time Signals ●●● 色● Effect of Sampling in the Frequency Domain Digital Processing of Sampling of Recovery of the Analog Signal Continuous-Time Signals Continuous-Time Signals Implication of the Sampling Process Sampling of Bandpass Signals 1.Introduction 1.Introduction 1.Introduction Digital processing of a continuous-time signal Conversion of a continuous-time signal into Since the A/D conversion takes a finite involves the following basic steps: digital form is carried out by an analog-to- amount of time,a sample-and-hold (S/H) (1)Conversion of the continuous-time signal digital(A/D)converter circuit is used to ensure that the analog signal into a discrete-time signal, .The reverse operation of converting a digital at the input of the A/D converter remains (2)Processing of the discrete-time signal. signal into a continuous-time signal is constant in amplitude until the conversion is complete to minimize the error in its (3)Conversion of the processed discrete-time performed by a digital-to-analog (D/A) converter representation signal back into a continuous-time signal To prevent aliasing,an analog anti-aliasing filter is employed before the S/H circuit
Chapter 4 Chapter 4 Digital Processing of Continuous-Time Signals Part A Sampling of Continuous-Time Signals Part A: Samp g lin of Continuous-Time Signals Introduction Sampling of Continuous Sampling of Continuous-Time Signals Time Signals Eff t f S li i th F D i Effect of S ampling in the Frequency requency Domain Recovery of the Analog Signal Implication of the Sampling Process Sampling of Sampling of Bandpass Bandpass Signals Signals 3 1. Introduction Digital processing of a continuous-time signal involves the following basic steps: involves the following basic steps: (1) Conversion of the continuous-time signal into a discrete-time signal, (2) Processing of the discrete (2) Processing of the discrete-time signal time signal, (3) Conversion of the processed discrete-time si l b k i t ti ignal back into a continuous-ti i l me signal 4 1. Introduction Conversion of a continuous-time signal into digital form is carried out by an digital form is carried out by an analog-todigital (A/D) converter The reverse operation of converting a digital signal into a continuous-time signal is performed by a digital-to-analog (D/A) converter 5 1. Introduction Since the A/D conversion takes a finite amount of time a amount of time, a sample-and-hold (S/H) hold (S/H) circuit is used to ensure that the analog signal at the input of the A/D converter remains at the input of the A/D converter remains constant in amplitude until the conversion is compl t t i i i th i it lete to minimize the error in its representation To prevent aliasing, an analog anti-aliasing aliasing filter is employed before the S/H circuit. 6 p y
1.Introduction 1.Introduction 1.Introduction S/H circuit often consists of a capacitor to An anti-aliasing filter is a filter used before a signal sampler,to restrict the bandwidth of a signal to To smooth the output signal of the D/A store the analogue voltage,and an electronic approximately satisfy the sampling theorem.Since converter,which has a staircase-like switch or gate to alternately connect and the theorem states that unumbiguous interpretation of waveform,an analog reconstruction filter is disconnect the capacitor from the analogue the signal from its samples is possible only when the used. input. power of frequencies outside the Nyquist handwidth is zero,the anti-aliasing filter would have to have The complete hlock-diagram is shown hlew perfect stop-band rejection to completely satisfy the theorem.Every realizable anti-aliasing filter will Anti-allasing S/H Deta Reconstruction D/A permit some aliasing to occur;the amount of aliasing that does occur depends on how good the filter is. 2.Sampling of Continuous-Time 2.Sampling of Continuous-Time 1.Introduction Signals Signals .Since both the anti-aliasing filter and the As indicated earlier.discrete-time signals in reconstruction filter are analog lowpass many applications are generated by sampling filters,we will review the theory behind the continuous-time signals design of such filters in this chapter It is obvious that identical discrete time Also.the most widely used IIR digital filter signals may result from the sampling of more design method is based on the conversion of than one distinct continuous-time function an analog lowpass prototype In fact.there exists an infinite number of continuous-time signals,which when sampled lead to the same discrete-time signal %
1. Introduction S/H circuit often consists of a capacitor to store the analogue voltage and an electronic store the analogue voltage, and an electronic switch or gate to alternately connect and di h i f h l disconnect the capacitor from the analogue input. 7 1. Introduction An anti-aliasing filter is a filter used before a signal samp, g ler to restrict the bandwidth of a signal to approximately satisfy the sampling theorem. Since the theorem states that unambiguous interpretation of th i l f i l i ibl l h h he signal from its samples is possible only when the power of frequencies outside the Nyquist bandwidth is zero the anti is zero, the anti-aliasing filter would have to have aliasing filter would have to have perfect stop-band rejection to completely satisfy the theorem Every realizable anti theorem. Every realizable anti-aliasing filter will aliasing filter will permit some aliasing to occur; the amount of aliasing that does occur depends on how good the filter is. 8 p g 1. Introduction To smooth the output signal of the D/A converter, whi h h i hich has a staircase-like waveform, an analog reconstruction filter is used. The complete block-diagram is shown blew Anti-aliasing S/H A/D D/A Digital Reconstruction filter S/H A/D D/A Processing filter 9 1. Introduction Since both the anti-aliasing filter and the reconstructi fil onter are anal l og owpass filters, we will review the theory behind the d i f h fil i hi h design of such filters in this chapter Also, yg the most widely used IIR digital filter design method is based on the conversion of an analog lowpass prototype 10 2. Sampling of Continuous-Time Signals As indicated earlier, discrete-time signals in many applications are generated by sampling many applications are generated by sampling continuous-time signals It is obvious that identical discrete time signals may result from the sampling of more than one distinct continuous-time function In fact there exists an infinite number of In fact, there exists an infinite number of continuous-time signals, which when sampled lead to the same discrete-time signal 11 lead to the same discrete-time signal 2. Sampling of Continuous-Time Signals 1 g1(t) 0.5 g1(n) g2(t) g2(n) g3(t) 0 mplitudeg3(n) -0.5 A 0 0.2 0.4 0.6 0.8 1 -1 Time 12
2.Sampling of Continuous-Time 2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency Signals Domain Domain However,under certain conditions,it is Let g(t)be a continuous-time signal that is G.()=[g.(d possible to relate a unique continuous-time sampled uniformly at t=n7,generating the The frequency-domain representation of g(n) signal to a given discrete-time signals sequence g(n)where g(n)=g(nT)with T being the sampling period is given by its discrete-time Fourier transform If these conditions hold,then it is possible to recover the original continuous-time signal The reciprocal of Tis called the sampling (DTFT): G(e)=g(ne from its sampled values frequency Fr,i.e.,F=1/T To establish the relation between G(j)and We next develop this correspondence and the Now,the frequency-domain representation of G(e),we treat the sampling operation associated conditions g(r)is given by its continuous-time Fourier mathematically as a multiplication of g()by transform(CTFT): a periodic impulse train p(r): 13 14 15 2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency Domain Domain Domain .There are two different forms ofG(j): .p(r)consists of a train of ideal impulses with a .gr)is a continuous-time signal consisting of period T as shown below a train of uniformly spaced impulses with the One form is given by the weighted sum of the impulse at t=nT weighted by the sampled 一 value g(nT)of g(t)at that instant CiFsf 2 To derive the second form,we note that p(r) The multiplication operation yields an can be expressed as a Fourier series: impulse train: ,()=g.(p()=g.(nT)6u-nT) p(t)= T正 where T 2π 17 T
2. Sampling of Continuous-Time Signals However, under certain conditions, it is possible to relate a unique continuous possible to relate a unique continuous-time signal to a given discrete-time signals If these conditions hold, then it is possible to recover the original continuous-time signal from its sampled values We next develop this correspondence and the We next develop this correspondence and the associated conditions 13 2.1 Effect of Sampg q y ling in the Frequency Domain Let ga(t) be a continuous-time signal that is sampled uniformly at sampled uniformly at t = nT, generating the generating the sequence g(n) where g(n)= ga(nT) with T being the being the sampling period sampling period The reciprocal of T is called the sampling frequency F i T ,.e., Now, the frequency-domain representation of 1/ FT T , qy p ga(t) is given by its continuous-time Fourier transform (CTFT): 14 transform (CTFT): 2.1 Effect of Sampg q y ling in the Frequency Domain ( ) () j t G j g t e dt a a The frequency-domain representation of g(n) is given by its discrete-time Fourier transform (DTFT): ( ) () j j n n Ge gne To establish the relation between and , we treat the sampling operation n ( ) G j a ( ) j G e , p gp mathematically as a multiplication of ga(t) by a periodic impulse train p(t): G e( ) 15 a pe od c pu se t a p(t): 2.1 Effect of Sampg q y ling in the Frequency Domain p(t) consists of a train of ideal impulses with a peri do T as sh bl owne ow (t) g (t) p(t) ga(t) p(t) gp(t) … … t ˉ2TˉT 0 T 2T The multiplication operation yields an impulse train: p(t) 2T T 0 T 2T impulse train: () () () ( ) ( ) pa a g t g t pt g nT t nT 16 n 2.1 Effect of Sampg q y ling in the Frequency Domain ga(t) is a continuous-time signal consisting of a trai f if l d i l i h h in of uniformly spaced impulses with the impulse at t = nT weighted by the sampled value g ( T) f ( ) h i a(nT) of ga(t) at that instant ga(t) gp(t) ga(t) 0 t 0 t 17 2.1 Effect of Sampg q y ling in the Frequency Domain There are two different forms of : One form is given by the weighted sum of the ( ) Gp j One form is given by the weighted sum of the CTFTs of : ( ) t nT () () j nT G j g nT e To derive the second form, we note that p(t) () () p a n G j g nT e , p( ) can be expressed as a Fourier series: 2 1 1 j kt j kt where 1 1 T j T j kt k k pt e e T T 2 T 18 T
2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency Domain Domain Domain The impulse train g(r)therefore can be expressed as .Therefore,G(j)is a periodic function of ·The frequency range-2,/2≤2≤2，/2is consisting of a sum of shifted and scaled called the baseband or Nyquist band replicas of G(j),shifted by integer The above result is more commonly known as From the frequency-shifting property of the multiples of and scaled by I/T the sampling theorem: CTFT,the CTFT ofg()is given by The term on the RHS of the previous equation .Letg (r)be a band-limited signal withG(j)=0 GU2-k2,》 for k=0 is the baseband portion of G(j), for >then g(r)is uniquely determined .Hence,an alternative form of the CTFT of and each of the remaining terms are the by its samples g(mT,-o≤n≤oo,if frequency translated portions of G(j) (m-号2aa-a》 is given by .2π 2,≥2 where0,=T 20 21 2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency Domain Domain Domain Illustration of the frequency-domain effects of time-domain sampling ·→If2,≥22.，g0 can be recovered The spectra of the filter and pertinent signals exactly from g(r)by passing it through an are shown below ideal lowpass filter H(j)with a gain Tand a cutoff frequency greater than and less than as shown below T14 23 0 品
2.1 Effect of Sampg q y ling in the Frequency Domain The impulse train gp(t) therefore can be expressed as 1 F hf hif i f h 1 () () T j kt p a k gt e gt T From the frequency-shifting property of the CTFT, the CTFT of is given by ( ) T j kt a e g t Hence an alternative form of the CTFT of ( ( )) Gj k a T Hence, an alternative form of the CTFT of is given by 1 Gj Gj k ( ) ( ( )) 19 ( ) ( ( )) p aT k Gj Gj k T 2.1 Effect of Sampg q y ling in the Frequency Domain Therefore, is a periodic function of i ti f f hift d d l d Gp ( ) j consisting of a sum of shifted and scaled replicas of , shifted by integer lti l f d l d b 1/T ( ) G j a multiples of and scaled by 1/T The term on the RHS of the previous equation T p q for k = 0 is the baseband baseband portion portion of , and each of the remaining terms are the ( ) Gp j g frequency translated portions of ( ) Gp j 20 2.1 Effect of Sampg q y ling in the Frequency Domain The frequency range is called the baseband or Nyquist band /2 /2 T T called the baseband or Nyquist band The above result is more commonly known as the sampling theorem: Let ga(t) be a band be a band-limited signal with limited signal withG j ( )0 for , then ga(t) is uniquely determined by its samples g (nT) if ( )0 G j a m by its samples ga(nT), , n if where n 2 T m 2 T T 21 2.1 Effect of Sampg q y ling in the Frequency Domain Illustration of the frequency-domain effects of time-domain sampling ( ) G j a 1 … … P( ) j 1 0 m m … … T 0 T 2T 3T ( ) G j p … T 0 T m T 2 m T m … 1/T T T T m ( ) G j p 1/T 22 … 0 T T 2 m m T … T 2 T 2.1 Effect of Sampg q y ling in the Frequency Domain If , ga 2 (t) can be recovered T m exactly from gp(t) by passing it through an ideal lowpass filter with a gain ( ) T and H j r a cutoff frequency greater than and less than as shown below r c m T m ga(t) gp(t) ( ) Hr j ˆ ( ) a g g t a( ) p(t) ( ) r j ( ) a g 23 2.1 Effect of Sampg q y ling in the Frequency Domain The spectra of the filter and pertinent signals are shown below are shown below ( ) G j p T m 1/T … T 0 T 2 m T m … H j r ( ) mc Tm T 0 c c ˆ ( ) G j a 1 24 0 m m
2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency Domain Domain Domain .On the other hand,if .20. Oversampling Hence,in compact disc(CD)music systems, or aliased into the baseband. When <20. Undersampling a sampling rate of 44.1 kHz,which is slightly 。Several terms: When Qr=20. Critical sampling higher than twice the signal bandwidth,is used Note:A pure sinusoid may not be recoverable from its critically sampled version 26 27 2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency Domain Domain Domain Example 2 These three transforms are plotted below These continuous-time signals sampled at a .Consider the three continuous time sinusoidal GU. rate of T=0.1 sec,i.e.,with a sampling signals: frequency =20 rad/sec gi(t)=cos(6m1),g(t)=cos(14mt),g;(r)=cos(26mt) 606 The sampling process generates the Their corresponding CTFTs are: G() continuous-time impulse trains,gi(),g2(1), and g3p(r) G(j2)=π[8(2-6π)+6(2+6π] G2(j2)=π[6Q-14x)+82+14z)] G,2 Their corresponding CTFTs are given by G3(Uj2)=π[82-26π)+82+26m)] G,U2=102Gj0-k2,,1s1s3
2.1 Effect of Sampg q y ling in the Frequency Domain On the other hand, if , due to the overlap of the shifted replicas of the 2 T m overlap of the shifted replicas of , G j ( ) the spectrum cannot be separated by filtering to recover because of the distortion ( ) G j a ( ) G j a filtering to recover because of the distortion caused by a part of the replicas i di t l t id th b b d f ld d b k ( ) G j a immediately outside the baseband folded back or aliased into the baseband. Several terms: 25 2.1 Effect of Sampg q y ling in the Frequency Domain Nyquist condition / 2 Folding frequency 2 T m Folding frequency Nyquist frequency / 2 T m Nyquist rate When Oversampling 2 m When 2 Oversampling When Undersampling 2 T m 2 T m When Critical sampling Note: A pure sinusoid may not be recoverable from its 2 T m 26 Note: A pure sinusoid may not be recoverable from its critically sampled version 2.1 Effect of Sampg q y ling in the Frequency Domain Example 1 In high-quality analog music signal processing, a bandwidth of 20 kHz has been determined to preserve the fidelity (؍ⵏᓖ) Hence in compact disc ( Hence, in compact disc (CD) music systems ) music systems, a sampling rate of 44.1 kHz, which is slightly higher than twice the signal bandwidth is higher than twice the signal bandwidth, is used 27 2.1 Effect of Sampg q y ling in the Frequency Domain Example 2 C id th th ti ti i id l Consider the three continuous time sinusoidal signals: Their corresponding CTFTs are: 12 3 gt t gt t gt t ( ) cos(6 ), ( ) cos(14 ), ( ) cos(26 ) Their corresponding CTFTs are: G1() ( j 6 ) ( 6 ) 1 2 () ( ) ( ) ( ) ( 14 ) ( 14 ) ( ) ( 26 ) ( 26 ) j G j G j 28 G j 3 ( ) ( 26 ) ( 26 ) 2.1 Effect of Sampg q y ling in the Frequency Domain These three transforms are plotted below G j 1( ) 6 0 6 2 G ( ) j 0 14 0 14 3 G ( ) j 29 0 26 26 2.1 Effect of Sampg q y ling in the Frequency Domain These continuous-time signals sampled at a rate of rate of T = 0 1 sec 0.1 sec, i e ith a sampling i.e., with a sampling frequency rad/sec 20 T The sampling process generates the continuous-time impulse trains, g1p(t), g2p p g (t), 1p( ) g2p ( ) and g3p (t) Their corresponding CTFTs are given by Their corresponding CTFTs are given by ( ) 10 ( ( )), 1 3 G j Gj k l lp l T 30 p k
2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency Domain Domain Domain .Plots of the 3 CTFTs are shown below We now derive the relation between the ·Observation:We have DTFT of g(n)and the CTFT of gp() 。To this end we compare G(e)=G,(jQ)- -6 0 6s G(e)=g(ne or,equivalently, G. with G(j)=G(e)ar From the above observation and G()- and make use of s. Cj=T2cUQ-k2,》 g(n)=g(nT)-oo<n<oo 31 2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency 2.1 Effect of Sampling in the Frequency Domain Domain Domain We arrive at the desired result given by The relation derived on the previous slide can .It follows that G(e)is obtained from G(j) be alternately expressed as .(- by applying the mapping =@/T oe-宁2cU- .Now,the CTFTG (j)is a periodic function of with a period =2/T 2c(号) form G(e)=G(j)a Because of the mapping,the DTFT G(el)is a periodic function of with a period 2 2(停) or from G()=G(e)
2.1 Effect of Sampg q y ling in the Frequency Domain Plots of the 3 CTFTs are shown below 1 ( ) G p j 1 H j ( ) ( ) p j 6 10 20 6 20 0.1 ( ) H j r c c Spectrum lines painted by Red and The cutoff frequency of the by Red 20 6 0 6 20 and Green colors designate aliases of the lowpass filter is chosen as 2 ( ) G p j 10 0 1 ( ) H j r aliases 10 c chosen as 0 20 14 14 20 0.1 6 6 c c 34 3 ( ) G p j 10 0.1 ( ) H j r 31 40 20 6 0 6 20 40 c c 26 26 2.1 Effect of Sampg q y ling in the Frequency Domain We now derive the relation between the DTFT f o g(n) and th CTFT f d the CTFT of g (t) p To this end we compare with ( ) () j j n n Ge gne with n () () j nT G j g nT e p a and make use of g n g nT n () ( ) n 32 () ( ) a g n g nT n 2.1 Effect of Sampg q y ling in the Frequency Domain Observation: We have or equivalently / () () j p T Ge G j or, equivalently, () () j p T G j Ge From the above observation and 1 Gj Gj k ( ) ( ( )) p aT k Gj Gj k T 33 2.1 Effect of Sampg q y ling in the Frequency Domain We arrive at the desired result given by 1 ( ) j G e G j jk a T T / 1 k T T G j jk a T k G j jk T T 1 2 a k k Gj j T TT 34 T TT k 2.1 Effect of Sampg q y ling in the Frequency Domain The relation derived on the previous slide can be alternately expressed as 1 ( ) j T G e G j jk form ( ) j a T k G e G j jk T () () j Ge G j form or from / () () j p T Ge G j () () j G j Ge or from () () j p T G j Ge 35 2.1 Effect of Sampg q y ling in the Frequency Domain It follows that is obtained from ( ) j G e ( ) Gp j by applying the mapping Now the CTFT is a periodic function ( ) ( ) p j /T Now, the CTFTG j ( ) is a periodic function of with a period B f h i h DTFT i ( ) G j p 2 / T T j Because of the mapping, the DTFT is a periodic function of with a period ( ) j G e 2 36
2.2 Recovery of the Analog Signal 2.2 Recovery of the Analog Signal 2.2 Recovery of the Analog Signal We now derive the expression for the output Thus,the impulse response is given by Therefore,the output (t)of the ideal g (r)of the ideal lowpass reconstruction lowpass filter is given by: filter H (j)as a function of the samples g(n) a0-上Umen-严m .The impulse response h(r)of the lowpass 或0=，0r60-立k-0 reconstruction filter is obtained by taking the =sin(1) -0≤1≤0 21/2 .Substituting h (r)=sin(1)/21/2 in the inverse DTFT of: above and assuming .=,/2=/T for T,2≤2 The input to the lowpass filter is the impulse H,U2)= train go(t) simplicity,we get 0,2>2 g,()=>g(m)o(-nT) 0=2g sinπ(t-nT)/T] 37 x(t-nT)/T 2.2 Recovery of the Analog Signal 2.2 Recovery of the Analog Signal 2.3 Implication of the Sampling Process It can be shown that when =,/2 in The relation g(rT)=g(r)=g(rT)holds Consider again the three continuous-time h(t)=sin2i whether or not the condition of the sampling signals:g(r)=cos(6mr).g.(r)=cos(14mt), Ω1/2 theorem is satisfied and g;(1)=cos(26mt) h,(0)=1 and h(nT)=0 for m0 However,g(r)=g.(r)for all values of t only .The plot of the CTFTG (j)of the sampled 0=2g物mt-nD1☐ ·As a result,,from if the sampling frequency satisfies the version of g(r)is shown below condition of the sampling theorem π(t-nT)/T G.( 有国一黄 we observe g(rT)=g(r)=g.(rT) for all integer values of r in the range<< -20.x -6r06x 40 41 42
2.2 Recovery of the Analog Signal We now derive the expression for the output f df sˆ ( )t of th id l l t ti f the ideal lowpass reconstruction filter as a function of the samples g(n) ( ) a g t ( ) H j r The impulse response hr(t) of the lowpass reconstruction filter is obtained by g takin the inverse DTFT of : T, ( ) 0, c r c T H j 37 2.2 Recovery of the Analog Signal Thus, the impulse response is given by 1 T () ( ) 2 2 cc jt jt r r T ht H j e d e d sin( ) , / 2 ct t t The input to the lowpass filter is the impulse t i (t) / 2 c t train gp(t) () ( ) ( ) p n g t g n t nT 38 n 2.2 Recovery of the Analog Signal Therefore, the output of the ideal l filt i i b ˆ ( ) a g t lowpass filter is given by: g t g t h t g n h t nT ˆ () () () ( ) ( ) Substituting in the () () () ( ) ( ) a pr r n g t g t h t g n h t nT Subs u g e ht t t ( ) sin( ) / / 2 above and assuming for simplicity, we get ( ) sin( ) / / 2 r cc ht t t /2 / c T T simplicity, we get sin[ ( ) / ] ˆ () ( ) t nT T g t gn 39 () ( ) ( )/ a n g t gn t nT T 2.2 Recovery of the Analog Signal It can be shown that when in sin t / 2 c T h (0) 1 d h ( ) 0 f 0 sin ( ) / 2c r T t h t t hr(0)=1 and hr(nT)=0 for n0 As a result, from sin[ ( ) / ] t nT T s a esu , o ˆ () ( ) ( )/ a n t nT T g t gn t nT T we observe for all integer values of r in the range ˆ ( ) () ( ) a a g rT g r g rT r 40 for all integer values of r in the range r 2.2 Recovery of the Analog Signal The relation holds h th t th diti f th li ˆ ( ) () ( ) a a g rT g r g rT whether or not the condition of the sampling theorem is satisfied However, for all values of t only if the sampling frequency satisfies the ˆ () () a a gt gt T pg q y condition of the sampling theorem 41 2.3 Implication of the Sampling Process Consider again the three continuous-time si lgna s: , , ( ) (6 ) ( ) (14 ) and 3 gt t ( ) cos(26 ) 2 gt t 1( ) cos(6 ) gt t ( ) cos(14 ) The plot of the CTFT of the sampled version of g1(t) is shown below 1 ( ) G p j g1( ) 1 ( ) G j p 10 ( ) H j r 0 20 6 6 20 0.1 c c 42 20 6 0 6 20
2.3 Implication of the Sampling Process 2.3 Implication of the Sampling Process 2.3 Implication of the Sampling Process .From the plot,it is apparent that we can recover any of its frequency-translated Likewise.we can recover the aliased There is no aliasing distortion unless the versions cos(20k±6)πoutside the baseband component cos(6t)from the original continuous-time signal also contains baseband by passing through an ideal analog sampled version of either g2p(r)or g(r)by the component cos(6/t) bandpass filter g()with a passband centered passing it through an ideal lowpass filter with Similarly,from either g2(r)or g3(t)we can at2=(20k±6)π a frequency response: recover any one of the frequency-translated For example,to recover the signal cos(34mr). 0.1,0≤2≤(6+④)x versions,including the parent continuous-time it will be necessary to employ a bandpass H(j)= 0,otherwise signal cos(14r)or cos(26nr)as the case may filter with a frequency response be,by employing suitable filters H(j)= 0.1,(34-△)z≤≤34+4)z otherwise A small n棉er 3.Sampling of Bandpass Signals 3.Sampling of Bandpass Signals 3.Sampling of Bandpass Signals The conditions developed earlier for the .There are applications where the continuous- However,due to the bandpass spectrum of the unique representation of a continuous-time time signal is bandlimited to a higher continuous-time signal,the spectrum of the signal by the discrete-time signal obtained by frequency range2z≤l2≤2 with2,>0 discrete-time signal obtained by sampling will uniform sampling assumed that the Such a signal is usually referred to as the have spectral gaps with no signal components continuous-time signal is bandlimited in the bandpass signal present in these gaps frequency range from dc to some frequency Such a continuous-time signal is commonly To prevent aliasing a bandpass signal can of Moreover,if is very large,the sampling course be sampled at a rate greater than twice rate also has to be very large which may not referred to as a lowpass signal the highest frequency,i.e.by ensuring be practical in some situations 2,222H 为 47
F h l ii h 2.3 Implication of the Sampling Process From the plot, it is apparent that we can recover any of its frequency-translated versions outside the baseband by passing through an ideal analog cos 20 6 k t ! " # bandpass filter g1p(t) with a passband centered at ! 20 6 k For example, to recover the signal cos(34pt), it will be necessary to employ a bandpass cos(34 ) t it will be necessary to employ a bandpass filter with a frequency response 0 1 (34 ) (34 ) $ $ 43 0.1, (34 ) (34 ) ( ) 0, otherwise H j r $ $ A small number 2.3 Implication of the Sampling Process Likewise, we can recover the aliased b bd t aseband component cos(6 ) ( t)6pt f th rome sampled version of either g2p(t) or g3p(t) by i it th h id l l filt ith cos(6 ) t passing it through an ideal lowpass filter with a frequency response: 0.1, 0 (6 ) ( ) 0 otherwise H j r $ 0, otherwise 44 2.3 Implication of the Sampling Process There is no aliasing distortion unless the ori i l ti iginal continuous-ti i l l t i time signal also contains the component cos(6 ) t Similarly, from either g2p(t) or g3p(t) we can recover any qy one of the frequency-translated versions, including the parent continuous-time signal cos(14t) or cos(26t) as the case may be, by employing suitable filters 45 3. Sampling of Bandpass Signals The conditions developed earlier for the uni t ti f ti ique representation of a continuous-time signal by the discrete-time signal obtained by unif li d th t th iform sampling assumed that the continuous-time signal is bandlimited in the f f dt f frequency range from dc to some frequency Such a continuous-time sig y nal is commonly referred to as a lowpass signal 46 3. Sampling of Bandpass Signals There are applications where the continuousti i l i b dli it d t hi h time signal is bandlimited to a higher frequency range a with L H 0 L Such a signal is usually referred to as the bandp g ass signal To prevent aliasing a bandpass signal can of course be sampled at a rate greater than twice course be sampled at a rate greater than twice the highest frequency, i.e. by ensuring 2 47 T H 3. Sampling of Bandpass Signals However, due to the bandpass spectrum of the continuous-ti i l th t f th time signal, the spectrum of the discrete-time signal obtained by sampling will h t l ith i l t have spectral gaps with no signal components present in these gaps Moreover, if is very large, the sampling rate also has to be very large which may not H yg y be practical in some situations 48
3.Sampling of Bandpass Signals 3.Sampling of Bandpass Signals 3.Sampling of Bandpass Signals .A more practical approach is to use under- We choose the sampling frequency to ·This leads toG,U2=T∑G.(jn-j2k(a2 sampling satisfy the condition 。Let△2-2a-2,define the bandwidth of the 22a 27=2(△2)= .As before,G (j)consists of a sum of G(j) M band中ass signal and replicas of G(j)shifted by integer which is smaller than 20,the Nyquist rate multiples of twice the bandwidth An and Assume first that the highest frequency Substitute the above expression for,in scaled by 1/T contained in the signal is an integer multiple The amount of shift for each value ofk of the bandwidth,i.e.. 1 ensures that there will be no overlap between 2a=M(△2) ..(- all shifted replicas no aliasing 49 50 5到 3.Sampling of Bandpass Signals 3.Sampling of Bandpass Signals Figure below illustrates the idea behind As can be seen,g(t)can be recovered from g()by passing it through an ideal bandpass G filter with a passband given byss and a gain of T Note:Any of the replicas in the lower G frequency bands can be retained by passing g(r)through bandpass filters with passbands 2z-k(AQ)≤2≤2a-k(A2),providing a translation to lower frequency ranges 1sk<M-I 52
3. Sampling of Bandpass Signals A more practical approach is to use undersampling Let define the bandwidth of the $ H L bandpass signal Assume first that the highest frequency H L Assume first that the highest frequency contained in the signal is an integer multiple of the bandwidth i e H of the bandwidth, i.e., ( ) $ H M 49 3. Sampling of Bandpass Signals We choose the sampling frequency to i f h di i T satisfy the condition 2 2( ) H T M $ which is smaller than , the 2 Nyquist rate H ( ) T M Substitute the above expression for in T 1 G j G j jk ( ) p ( ) a T k G j G j jk T 50 3. Sampling of Bandpass Signals This leads to 1 ( ) 2( ) G j G j jk p a T $ As before, consists of a sum of and replicas of shifted by integer T k ( ) Gp j ( ) G j a and replicas of G j ( ) shifted by integer multiples of twice the bandwidth DWand scaled by 1/T ( ) G j a $ scaled by 1/T The amount of shift for each value of k ensures th h ill b l b hat there will be no overlap between all shifted replicas no aliasing 51 3. Sampling of Bandpass Signals Figure below illustrates the idea behind ( ) G j a 0 H L L H ( ) G j p 0 H L H L 52 3. Sampling of Bandpass Signals As can be seen, ga(t) can be recovered from g (t) by passing it through an ideal bandpass p(t) by passing it through an ideal bandpass filter with a passband given by and a gain of T L H and a gain of T Note: Any of the replicas in the lower f b d b t i db i frequency bands can be retained by passing gp(t) through bandpass filters with passbands , providing a translation to lower frequency ranges () () L H $ $ k k 1 1 k M 53