PDF Digital Processing of

(1)

1

Digital Processing of Continuous

Continuous- -Time Signals Time Signals

• Digital processing of a continuous-time signal involves the following basic steps:

(1) Conversion of the continuous-time signal into a discrete-time signal,

(2) Processing of the discrete-time signal, (3) Conversion of the processed discrete- time signal back into a continuous-time signal

2

Digital Processing of Digital Processing of Continuous

• Conversion of a continuous-time signal into digital form is carried out by ananalog-to- digital(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

3

Digital Processing of Digital Processing of Continuous

Continuous- -Time Signals Time Signals

• Since the A/D conversion takes a finite amount of time, asample-and-hold(S/H) circuitis used to ensure that the analog signal at the input of the A/D converter remains constant in amplitude until the conversion is complete to minimize the error in its representation

4

Digital Processing of Digital Processing of Continuos

Continuos- -Time Signals Time Signals

• To prevent aliasing, an analoganti-aliasing filteris employed before the S/H circuit

• To smooth the output signal of the D/A converter, which has a staircase-like waveform, an analogreconstruction filter is used

5

Digital Processing of Digital Processing of Continuous

Complete block-diagram

• Since both the anti-aliasing filter and the reconstruction filter are analog lowpass filters, we review first the theory behind the design of such filters

• Also, the most widely used IIR digital filter design method is based on the conversion of an analog lowpass prototype

Anti- aliasing

filter S/H A/D _processor^Digital D/A Reconstruction filter

6

Sampling of Continuous

Sampling of Continuous- -Time Time Signals

Signals

• As indicated earlier, discrete-time signals in many applications are generated by

sampling continuous-time signals

• We have seen earlier that identical discrete- time signals may result from the sampling of more than one distinct continuous-time function

(2)

7

Sampling of Continuous

Signals

• In fact, there exists an infinite number of continuous-time signals, which when sampled lead to the same discrete-time signal

• However, under certain conditions, it is possible to relate a unique continuous-time signal to a given discrete-time signal

8

Sampling of Continuous

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 associated conditions

9

Effect of Sampling in the Effect of Sampling in the

Frequency Domain Frequency Domain

• Let be a continuous-time signal that is sampled uniformly at t= nT, generating the sequence g[n]where

with Tbeing the sampling period

• The reciprocal of Tis called the sampling frequency , i.e.,

∞

<

∞

−

=g nT n

n

g[ ] _a( ), )

(t g_a

T T

F =¹ FT

10

Frequency Domain Frequency Domain

• Now, the frequency-domain representation of is given by its continuos-time Fourier transform (CTFT):

• The frequency-domain representation of g[n]

is given by its discrete-time Fourier transform (DTFT):

) g_a(t

dt e t g j

G_a( Ω)=

∫

−^∞∞ _a() −^jΩ^t

∑

^∞=−∞ − ω

ω = _n ^j ⁿ

j g n e

e

G( ) [ ]

11

Effect of Sampling in the Effect of Sampling in the

• To establish the relation between and , we treat the sampling operation mathematically as a multiplication of by aperiodic impulse trainp(t):

∑δ −

= ^∞

−∞

= n

nT t t

p() ( )

) (e^j^ω G

) (jΩ G_a

×

) (t

g_a g_p(t)

) (t p

) (t g_a

12

• p(t) consists of a train of ideal impulses with a periodTas shown below

• The multiplication operation yields an impulse train:

∑ δ −

=

= ^∞

−∞

=

n a

a

p t g t p t g nT t nT

g () () () ( ) ( )

(3)

13

Effect of Sampling in the Effect of Sampling in the

• is a continuous-time signal consisting of a train of uniformly spaced impulses with the impulse att= nTweighted by the sampled value of at that instant

) (t g_p

) g_a(nT g_a(t)

14

Effect of Sampling in the Frequency Domain Frequency Domain

• There are two different forms of :

• One form is given by the weighted sum of the CTFTs of :

• To derive the second form, we make use of the Poisson’s formula:

where and is the CTFT of

) (jΩ G_p

) (t−nT δ

∑

^∞=−∞ − Ω

=

Ω _n _a ^j ^nT

p j g nT e

G ( ) ( )

T =2π/T Ω

t jk

k T

n t nT T ^∞₌_−∞ jk e ^Ω^T

∞=−∞ = ∑ Φ Ω

∑ φ( + ) ¹ ( )

) ( Ω Φ j )

φ(t

15

Effect of Sampling in the Effect of Sampling in the

• Fort= 0 reduces to

• Now, from the frequency shifting property of the CTFT, the CTFT of is given by

t jk

k T

n t nT T ^∞₌_−∞ jk e ^Ω^T

∞=−∞ = ∑ Φ Ω

∑ φ( + ) ¹ ( )

) ( )

( = ∑ Φ Ω

∑^∞₌_−∞φ ^∞_k₌_−∞ _T

n nT T¹ jk

t j at e g () ⁻ ^Ψ ))

( (j Ω+Ψ G_a

16

• Substituting in we arrive at

• By replacing with Ωin the above equation we arrive at the alternative form of the CTFT of

∑ Ω +Ψ

∑^∞n₌_−∞ ⁻ ^Ψ =T ^∞_k₌_−∞ a T nT

j

a nT e G j k

g ( ) ¹ ( ( ))

) ( )

( = ∑ Φ Ω

∑^∞₌_−∞φ ^∞_k₌_−∞ _T

n nT T¹ jk

Ψ

t j at e g

t = ⁻ ^Ψ

φ() ()

) (t g_p ) (jΩ G_p

17

Effect of Sampling in the Effect of Sampling in the

• The alternative form of the CTFT of is given by

• Therefore, is a periodic function of Ωconsisting of a sum of shifted and scaled replicas of , shifted by integer multiples of and scaled by

( )

∑

^∞

−∞

=

Ω

− Ω

= Ω

k

T T a

p j G j k

G ( ) ¹ ( )

) (jΩ G_p

) (jΩ G_a

ΩT

T 1

) g_p(t

18

• The term on the RHS of the previous equation fork= 0 is the basebandportion of , and each of the remaining terms are the frequency translated portions of

• The frequency range

• is called thebasebandorNyquist band )

(jΩ G_p

) (jΩ G_p

2 2

T

T ≤ Ω

Ω Ω≤

−

(4)

19

Effect of Sampling in the Effect of Sampling in the

• Assume is a band-limited signal with a CTFT as shown below

• The spectrum of p(t)having a sampling period is indicated below

) (t g_a

) (jΩ G_a

T

T =)_Ω²^π (jΩ P

20

• Two possible spectra of are shown below

) (jΩ G_p

21

Effect of Sampling in the Effect of Sampling in the

• It is evident from the top figure on the previous slide that if , there is no overlap between the shifted replicas of generating

• On the other hand, as indicated by the figure on the bottom, if , there is an overlap of the spectra of the shifted replicas of generating

) (jΩ G_a )

(jΩ G_p

) (jΩ G_p )

(jΩ G_a

m T > Ω Ω 2

m T < Ω Ω 2

22

Frequency Domain Frequency Domain

If , can be recovered exactly from by passing it through an ideal lowpass filter with a gain Tand a cutoff frequency greater than and less than as shown below

m T > Ω Ω 2 g_a(t)

) (t g_p

) (jΩ H_r

Ωc

Ωm Ω_T −Ω_m

23

• The spectra of the filter and pertinent signals are shown below

24

Effect of Sampling in the Effect of Sampling in the

• On the other hand, if , due to the overlap of the shifted replicas of , the spectrum cannot be separated by filtering to recover because of the distortion caused by a part of the replicas immediately outside the baseband folded back oraliasedinto the baseband

m T < Ω Ω 2

) (jΩ G_a )

(jΩ G_a

) (jΩ G_a

(5)

25

Effect of Sampling in the Frequency Domain Frequency Domain

Sampling theorem-Let be a bandlimited signal with CTFT for

• Then is uniquely determined by its samples , if

where

) (t g_a

0 ) (jΩ = G_a Ωm

>

Ω

) (t g_a

) (nT

g_a −∞≤n≤∞

m T ≥ Ω Ω 2

T =2π/T Ω

26

Effect of Sampling in the Effect of Sampling in the

• The condition is often referred to as the Nyquist condition

• The frequency is usually referred to as thefolding frequency²

ΩT m T ≥ Ω Ω 2

27

• Given , we can recover exactly by generating an impulse train

and then passing it through an ideal lowpass filter with a gain Tand a cutoff frequency satisfying

∑

^∞=−∞ δ −

= _n _a

p t g nT t nT

g () ( ) ( )

)}

(

{g_a nT g_a(t)

) (jΩ H_r

Ωc

(

_T _m

)

c

m<Ω < Ω −Ω Ω

28

Effect of Sampling in the Effect of Sampling in the

• The highest frequency contained in is usually called theNyquist frequency since it determines the minimum sampling frequency that must be used to fully recover from its sampled version

• The frequency is called the Nyquist rate

Ωm

m T = Ω Ω 2

) (t g_a

Ωm

2

) (t g_a

29

Frequency Domain Frequency Domain

• Oversampling-The sampling frequency is higher than the Nyquist rate

• Undersampling-The sampling frequency is lower than the Nyquist rate

• Critical sampling-The sampling frequency is equal to the Nyquist rate

• Note:A pure sinusoid may not be recoverable from its critically sampled version

30

Effect of Sampling in the Effect of Sampling in the

• In digital telephony, a3.4 kHz signal bandwidth is acceptable for telephone conversation

• Here, a sampling rate of8 kHz, which is greater than twice the signal bandwidth, is used

(6)

31

• In high-quality analog music signal

processing, a bandwidth of20 kHz has been determined to preserve the fidelity

• Hence, in compact disc (CD) music

systems, a sampling rate of44.1 kHz, which is slightly higher than twice the signal bandwidth, is used

32

Effect of Sampling in the Effect of Sampling in the

• Example -Consider the three continuous- time sinusoidal signals:

• Their corresponding CTFTs are:

) 6 cos(

)

1(t t

g = π

) 14 cos(

)

2(t t

g = π

) 26 cos(

)

3(t t

g = π

)]

26 ( ) 26 ( [ )

3(jΩ =πδ Ω− π +δΩ+ π G

)]

14 ( ) 14 ( [ )

2(jΩ =πδΩ− π +δ Ω+ π G

)]

6 ( ) 6 ( [ )

1(jΩ =πδΩ− π +δ Ω+ π G

33

• These three transforms are plotted below

34

Effect of Sampling in the Effect of Sampling in the

• These continuous-time signals sampled at a rate ofT= 0.1 sec, i.e., with a sampling frequency rad/sec

• The sampling process generates the

continuous-time impulse trains, , , and

• Their corresponding CTFTs are given by π

= Ω_T 20

)

1 (t g _p )

2 (t

g _p g₃_p(t)

(

( )

)

, 1 3

10 )

( Ω =

∑

^∞₌_−∞ _l Ω− Ω ≤l≤

lp j k G j k T

G

35

Frequency Domain Frequency Domain

• Plots of the 3CTFTs are shown below

36

• These figures also indicate by dotted lines the frequency response of an ideal lowpass filter with a cutoff at and a gainT= 0.1

• The CTFTs of the lowpass filter output are also shown in these three figures

• In the case of , the sampling rate satisfies the Nyquist condition, hence no aliasing

π

= Ω

=

Ω_c _T/2 10

)

1(t g

(7)

37

Effect of Sampling in the Effect of Sampling in the

• Moreover, the reconstructed output is precisely the original continuous-time signal

• In the other two cases, the sampling rate does not satisfy the Nyquist condition, resulting in aliasing and the filter outputs are all equal tocos(6πt)

38

Effect of Sampling in the Frequency Domain Frequency Domain

• Note:In the figure below, the impulse appearing atΩ= 6πin the positive

frequency passband of the filter results from the aliasing of the impulse in at

• Likewise, the impulse appearing atΩ= 6π in the positive frequency passband of the filter results from the aliasing of the impulse in at

)

2(jΩ π G

−

= Ω 14

π

= Ω 26 )

3(jΩ G

39

Frequency Domain Frequency Domain

• We now derive the relation between the DTFT ofg[n] and the CTFT of

• To this end we compare with

and make use of

) (t g_p

∑

^∞=−∞ − ω

ω = _n ^j ⁿ

j g n e

e

G( ) [ ]

∑

^∞=−∞ − Ω

=

Ω _n _a ^j ^nT

p j g nT e

G ( ) ( )

∞

<

∞

−

=g nT n n

g[ ] _a( ),

40

• Observation:We have

or, equivalently,

• From the above observation and

p T

j G j

e

G( ^ω)= ( Ω)_Ω₌_ω/

T p j Gej

G ( Ω)= ( ω)ω=Ω

( )

∑

^∞

−∞

= Ω− Ω

= Ω

k

T T a

p j G j k

G ( ) ¹ ( )

41

Effect of Sampling in the Effect of Sampling in the

we arrive at the desired result given by

k a T T

T

j G j jk

e G

/

1 ( )

) (

ω

= Ω

∞

−∞

=

ω = ∑ Ω− Ω

)

1 ∑ ( − Ω

= ^∞

−∞

=

ω

k a T T

T G j jk

)

( ²

1 ∑ −

= ^∞

−∞

=

π ω

k T

k a T

T G j j

42

• The relation derived on the previous slide can be alternately expressed as

• From or from

it follows that is obtained from by applying the mapping

∑ Ω− Ω

= ^∞₌_−∞

Ω

k a T

T T

j G j jk

e

G( ) ¹ ( )

p T

j G j

e

G( ^ω)= ( Ω)_Ω₌_ω/ T p j G ej

G ( Ω)= ( ω)ω=Ω

T

=ω

Ω G_p(jΩ) )

(e^j^ω G

(8)

43

Effect of Sampling in the Effect of Sampling in the

• Now, the CTFT is a periodic function ofΩwith a period

• Because of the mapping, the DTFT is a periodic function ofωwith a period2π

) (jΩ G_p

) (e^j^ω G

T =2π/T Ω

44

Recovery of the Analog Signal Recovery of the Analog Signal

• We now derive the expression for the output of the ideal lowpass reconstruction filter as a function of the samples g[n]

• The impulse response of the lowpass reconstruction filter is obtained by taking the inverse DTFT of :

) (jΩ H_r ) (t g^{^}_a

) (jΩ H_r

⎩⎨⎧

= Ω) (j H_r

c

T c

Ω

>

Ω Ω

≤ Ω , 0

, ) (t h_r

45

Recovery of the Analog Signal Recovery of the Analog Signal

• Thus, the impulse response is given by

• The input to the lowpass filter is the impulse train :

∫

−^ΩΩ Ω

Ω π

∞

π − Ω Ω= Ω

= ^c

ce d

d e j H t

h_r _r ^j ^t ^T ^j ^t

2 2

1 ( )

) (

∞

≤

∞ Ω −

= Ω t

t t

T c ,

2 /

) sin(

∑

^∞=−∞ δ −

= _n

p t g n t nT

g () [ ] ( )

) (t g_p

46

• Therefore, the output of the ideal lowpass filter is given by:

• Substituting in the above and assuming for simplicity

, we get ) (t g^{^}_a

∑

^∞

−∞

=

−

=

n

r p

r

a t h t g t g nh t nT

g^{^} () ()* () [ ] ( ) ) 2 / /(

) sin(

)

(t t t

h_r = Ω_c Ω_T

T T

c=Ω /2=π/ Ω

) (t g^{^}_a

∑

^∞

−∞

= π −

−

= π

n t nT T

T nT n t

g ( )/

] / ) ( ]sin[

[

47

Recovery of the Analog Signal Recovery of the Analog Signal

• The ideal bandlimited interpolation process is illustrated below

48

• It can be shown that when in

and for

• As a result, from we observe

for all integer values of rin the range

2 /

)

) sin(

( t

t

r T

t c

h Ω

= Ω

2

T/

c=Ω Ω 1

) 0

( =

hr h_r(nT)=0 n≠0 )

(t

g^{^}_a

∑

^∞=−∞ π −

−

= _n π

T nT t

T nT

n t

g ( )/

] / ) (

]sin[

[

) ( ] [ )

(rT g r g rT

g_a = = _a

∞

<

∞

− r

(9)

49

Recovery of the Analog Signal Recovery of the Analog Signal

• The relation

holds whether or not the condition of the sampling theorem is satisfied

• However, for all values of tonly if the sampling frequency satisfies the condition of the sampling theorem

) ( ] [ )

(rT g r g rT g^{^}_a = = _a

) ( ) (rT g rT g^{^}_a = _a

ΩT

50

Implication of the Sampling Process

Process

• Consider again the three continuous-time signals: , , and

• The plot of the CTFT of the sampled version of is shown below

) 6 cos(

)

1(t t

g = π g₂(t)=cos(14πt) )

26 cos(

)

3(t t

g = π

)

1(t g )

1 (t g_p

)

1 (jΩ G_p

51

Implication of the Sampling Implication of the Sampling

Process 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 bandpass filter with a passband centered at

] ) 6 20 cos[( k± πt

)

1 (t g_p

π

±

=

Ω (20k 6)

52

Implication of the Sampling Implication of the Sampling

Process Process

• For example, to recover the signalcos(34πt), it will be necessary to employ a bandpass filter with a frequency response

where∆is a small number

⎩⎨⎧

= Ω) (j H_r

otherwise ,

0

) 34 ( )

34 ( , 1 .

0 −∆ π≤Ω≤ +∆ π

53

Implication of the Sampling Implication of the Sampling

Process Process

• Likewise, we can recover the aliased baseband componentcos(6πt)from the sampled version of either or by passing it through an ideal lowpass filter with a frequency response:

⎩⎨⎧

= Ω) (j H_r

otherwise ,

0

) 6 ( )

6 ( , 1 .

0 −∆π≤Ω≤ +∆ π

)

2 (t

g _p g₃_p(t)

54

Implication of the Sampling Implication of the Sampling

Process Process

• There is no aliasing distortion unless the original continuous-time signal also contains the componentcos(6πt)

• Similarly, from either or we can recover any one of the frequency- translated versions, including the parent continuous-time signal or as the case may be, by employing suitable filters

)

2 (t

g _p g₃_p(t)

)

3(t g )

2(t g

(10)

55

Sampling of

Sampling of Bandpass Bandpass Signals Signals

• The conditions developed earlier for the unique representation of a continuous-time signal by the discrete-time signal obtained by uniform sampling assumed that the continuous-time signal is bandlimited in the frequency range from dc to some frequency

• Such a continuous-time signal is commonly referred to as a lowpass signal

Ωm

56

Sampling of

• There are applications where the continuous- time signal is bandlimited to a higher frequency range with

• Such a signal is usually referred to as the bandpass signal

• To prevent aliasing a bandpass signal can of course be sampled at a rate greater than twice the highest frequency, i.e. by ensuring

H T ≥ Ω Ω 2

H L≤Ω≤Ω

Ω Ω_L>0

57

Sampling of

• However, due to the bandpass spectrum of the continuous-time signal, the spectrum of the discrete-time signal obtained by sampling will 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 be practical in some situations

ΩH

58

Sampling of

• A more practical approach is to use undersampling

• Let define the bandwidth of the bandpass signal

• Assume first that the highest frequency contained in the signal is an integer multiple of the bandwidth, i.e.,

ΩH L

H−Ω Ω

=

∆Ω

(∆Ω)

= Ω_H M

59

Sampling of

• We choose the sampling frequency to satisfy the condition

which is smaller than , the Nyquist rate

• Substitute the above expression for in ΩH

2

ΩT

T M^H

= Ω

∆Ω

=

Ω 2( ) ²

ΩT

( )

∑

^∞

−∞

=

Ω

− Ω

= Ω

k

T T a

p j G j k

G ( ) ¹ ( )

60

Sampling of

• This leads to

• As before, consists of a sum of and replicas of shifted by integer multiples of twice the bandwidth∆Ωand scaled by1/T

• The amount of shift for each value ofk ensures that there will be no overlap between all shifted replicas no aliasing

( )

∑∞ −∞= Ω− ∆Ω

=

Ω k G j j k

j

G _a

p( ) T¹ 2 ( )

) (jΩ G_p

) (jΩ G_a

(11)

61

Sampling of

• Figure below illustrate the idea behind

) (jΩ G_a

0 Ω ΩL

H − Ω

− Ω_L Ω_H

) (jΩ G_p

0 Ω ΩL

H − Ω

− Ω_L Ω_H

62

Sampling of Signals

• As can be seen, can be recovered from by passing it through an ideal

bandpass filter with a passband given by and a gain ofT

• Note:Any of the replicas in the lower frequency bands can be retained by passing

through bandpass filters with

passbands ,

providing a translation to lower frequency ranges

) (t g_a )

(t g_p

H L≤Ω≤Ω Ω

) (t g_p

) ( )

(∆Ω ≤Ω≤Ω − ∆Ω

−

Ω_L k _H k

1 1≤k≤M−