Lecture 41 Practical sampling and reconstruction.

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Lecture 4

Practical sampling and reconstruction

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Outline

Practical sampling– Aperture effect– Non ideal filters– Non-band limited input signals

Practical reconstruction Practical digital systems Discrete time Fourier transform

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Practical sampling

Practical sampling differs from ideal in the following respects– The sample (impulse) train actually consists of

pulses of duration – Real signal are time limited, therefore cannot

be band limited (the uncertainty principle of Fourier transform)

– Reconstruction filters are not ideal

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xa(t)

n=

s(t) = (tnT)

xs(t) = xa(nT)(tnT)n=

sample andhold filter

1h(t)

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Xa(j)1

s>2

1/T

/T

|Hs(j)|

/ /

H (j) = sin (/2) e-j/2

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s>2

|Xs(j)|/T

/T

Xs(j) = H(j) . 1T Xa(jjks )k= -

If / there is no significant distortion over signal band (otherwise equalization can compensate distortion)

Aperture effect

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Non ideal filters

The problem of non-ideal filtering can be combatted by increasing 1/T

Effectively, we are only using the middle portion of the filter, where it is closer to perfect

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Non-band limited signals

We may be only interested in a low frequency portion of a wide-band signal, e.g. speech only needs upto 3-4 Khz

There may be high frequency, wideband additive noise in the input signal

So we prefilter |c 1

0, |Haa(j) =

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Anti-aliasing filter Filter prior to sampling removes higher

frequency components which could have been moved into the lower frequency range by aliasing

Of course, we are distorting the signal, but this is in a frequency range in which we are not interested

If we were, we would need to use a higher sampling frequency

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Practical D/A conversion

We don’t have perfect interpolation Sample and hold The impulse response of a sample and hold filter is h(t)

1

h(t)

T

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Frequency domain

We need to compensate by adding a compensated reconstruction filter after the sample and hold process

|Hs(j)|

/ /

Ho(j) = sin (/2) e-j/2

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|Hr(j)|

/

e j/2, |

sin (/2)

~

Compensated reconstruction filter

0, |

~

Hr(j) =

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Ideal system

A/D D/Ah(n)

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Practical system

h(n)A/D

T

Antialiasing pre-filter

Sample and Hold

TSample and Hold

CompensatedReconstructionFilter

Haa(j)

D/A

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Effective frequency response

Heff(j) = Hr(j) H0(j) H(e jT) Haa(j)

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xa(t)

n=

S(t) = (tnT)

xs(t) = xa(nT)(tnT)n=

convert todiscretesequence x[n] = xa(nT)

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Back to sampling

Let xa(t)aperiodic

xs(t) = xa(nT)(tnT) aperiodic

Xs(j) = xa(nT)(tnT) e -

jtdt

n=

n=

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Xs(j) = xa(nT)(tnT) e -

jtdt

xa(nT) e -jnT

= x[n] e -jn

where =T X(e j) = x[n] e -jn is defined as the

discrete time Fourier transform of x[n]

n=

n=

n=

n=

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From the sampling theorem,

Xs(j) = Xa(jkjs)

thus,

X(e j) = Xa(jj )

In other words, on going from the continuous domain to the discrete domain, we undergo a scaling or normalization in frequency from =s to = 2

There is also a corresponding time normalization from t=T to n=1

n=

1T

k=

1T

T

2k T

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Discrete time Fourier transform

X(e j) = x[n] e -jn

– continuous and periodic with period 2

x[n] = X(e j) e jnd– discrete and aperiodic

n=

1

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Example

Let x[n] = u[n]-u[n-M]

= [n-k]

Then X(e j) = x[n] e -jn=e -jn

= = e -jM-1)/2

k=0

M-1

n=0

M-1

n=

1-e-jM

1-e-j

sin(M/2)

sin(/2)

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sin(M/2)

sin(/2)|X(e j)| =

/ / 0......

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Reading

Discrete time signals: Sections Sampling: Sections 8.2 Reconstruction, quantization, coding:

Sections 8.2 - p.363 Practical sampling and reconstruction:

Sections 8.2 - p.355 Fourier transform: Chapter 4