Sampling and ReconstructionEE 442 – Spring Semester

1

Lecture 8

Ideal sampling

Flat-top sampling

Natural sampling

Reading: Chapter 5; Section 5.2 Sampling – pp. 237 to 242

https://sipdtdevelopers.wordpress.com/tag/sampling/

2

Practical Sampling Process (Uniform Sample Interval)

Sample, Quantize & Encode

https://en.wikipedia.org/wiki/Audio_bit_depth

Red line: analog signal

Sampling time points

Allowedvalues

16 discrete levels shown

g(t) or m(t)

3

Remember this basic principle

https://www.quora.com/What-is-the-difference-between-Nyquist’s-signalling-theorem-and-Shannon’s-sampling-theorem

An absolutely band-limited waveform can’t be absolutely time-limited,

and

an absolutely time-limited waveform can’t be absolutely band-limited.

ft

Ш(t) = ( ) ( )S Sn n

t nT t nT

=− =−

− = +

12

12

( ) 1

n

n

t dt

+

−

= Ш

Shah function Ш(t): (TS is the sampling interval or period)

Using The “Fourier Transform” To Construct a “Sampling Function”

4

n is an integer

Shah Function (aka “Dirac Comb Function” or “Sampling Function”)

Sf 2 SfSf−2 Sf− 0 fST 2 STST−2 ST− 0 t

SPeriod T=

1S

S

fT

=Ш(t) Ш(f)

1

ST

Sampling Theorem

Sampling Theorem: A band-limited signal g(t) of bandwidth B (Hz)can be reconstructed exactly from data samples taken at a samplingrate fS if fS is greater than or equal to 2B (Hz).

Ideal sampling exists only mathematically on paper – it is achieved by multiplying by an impulse train. The unit impulse train (aka as Shah Function, or Dirac comb function, or sampling function) is

with interval TS seconds. Sampling signal g(t) at a uniform rate (withsampling period TS and sampling rate fS =(1/TS) ) yields

( ) ( )Sn

t t nT

=−

= −

( ) ( ) ( ) ( ) ( ) ( )S S Sn n

g t t g t t nT g nT t nT

=− =−

= − = − Ш

Ш

5

Fourier Transform of a Sampled Signal

The impulse train is a periodic signal of period TS and can be expressedas an exponential Fourier series, hence,

The Fourier transform of g(t) multiplied by Ш(t) is

Based upon the frequency-shifting property of the Fourier transform,the nth term is shifted by frequency nfS. We write this as

1 2( ) with 2S S

nS S

Sjn tt f

T Te

=−

= = =Ш

21

( ) ( ) ( )nS

Sjn f tF g t t g t

Te

=−

= Ш

1

( ) ( ) )

where ( ) ( )

( SnS

F g t tT

g t G f

G f nf

=−

=

−Ш

6

Fourier Transform of Sampled Signal (in pictures)

7After Lathi & Ding, 4th ed., 2009; p. 303.

( ) ( ( ))g t or m t

Ш(t)

FT

FT

G(f) consists of G(f), scaled by theconstant 1/TS , repeated periodicallywith period fS = (1/TS) as shown here.

G(f)g(t)

Baseband“band-limited”

spectra ofbandwidth B

G(f)

t

t

TS

Multiplespectra

ImpulseSampling

8

Why Multiple Spectrums Result From Sampling

The samples of two sine waves can be identical when at least one of them is at a frequency greater than half the sample rate.

http://www.wikiwand.com/en/Nyquist–Shannon_sampling_theorem

Lowest frequency

9

http://www.wikiwand.com/en/Nyquist–Shannon_sampling_theorem

Sampling Exactly at the Nyquist (Critical) Rate

A family of sinusoids at the critical frequency, all having the same sample sequences of alternating +1 and –1. That is, they all are aliases of each other, even though their frequency is not above half the sample rate.

Single SinusoidalWaveforms

Results innonuniquesolutions

10

Recall the Fourier Transform of a Single Pulse

1( )g t

t t

2( )g t

3( )g t

t

1( )G f

2( )G f

3( )G f

f

f

f

1

2

T1

2

T− 2

2

T2

2

T−

3

2

T3

2

T−

2T

1T

3T

1

1

T1

1

T−

2

1

T2

1

T

−

3

1

T3

1

T−

T1 > T2 > T3

Narrower spectrum corresponds to wider pulse.

11

Remember: Fourier Series for a Periodic Pulse Train

/2/2− TT− t

f

(1/ )Sf T=

(1/ )

(2/ )( 1/ )−

( 2 / )−

0

0

( )g t

( )G f

For a pulse train we have discrete frequencies (rather than a continuum of frequencies) spaced as (1/T) from each other. Thus, we have a combof sinusoids at . . . -2fS, -fS, 0, fS, 2fS, . . . , with amplitudes given by the sinc function associated with pulses of width .

sinc function envelope

12

Fourier Transform of Sampled Signal (in pictures)

( )g t

Ш(t)

FT

FT

G(f)g(t)

Band-limitedspectra of

bandwidth B

G(f)

t

t

The envelope of G(f) is the sinc function.

At f = 0 theamplitude

is A/TS

Important Question: Can g(t) be reconstructed from samples?

Answer: Yes, assuming no overlap meaning that fS > 2B, so thethe sampling interval TS must be < (1/2B).

Minimum Sampling Rate:The Minimum Sampling Rate fS = 2B is the Nyquist rate.Oversampling occurs when the rate exceeds the Nyquist rate.

Interpolation:Signal reconstruction is called Interpolation. We can recover g(t) by sending the samples through a band-limited filter ofbandwidth B (Hz).

How we do this is illustrated in the next slide . . .

13

Ideal Interpolation (Illustrated Diagrammatically)

14After Lathi & Ding, 4th ed., 2009; p. 305.

Idealband-limited

filter

FT

g(t) g(t)

H(f)h(t)

t

t

f

Note: Anoncausalimpulse response

Pass the samples through band-limited filter:

1

2B

1

2B

1

2B

−

A collectionof sinc (t)functions

Unit impulse

response h(t)

Ideal Interpolation (Shown Mathematically)

= ( ) 2 sinc(2 )Sh t BT Bt

At the Nyquist rate, with 2BTS = 1, then h(t) becomes

=( ) sinc(2 )h t Bt

( ) ( ) ( )

( ) ( ) sinc 2 ( )

S Sk

S Sk

g t g kT h t kT

g t g kT B t kT

= −

= −

Mathematics for the previous slide in reconstruction of g(t):

15

Practical Reconstruction For Non-ideal Samples – I

Impulse functions are not physically realizable, so a sampling pulse can not be truly instantaneous. Here p(t) is the reconstruction pulse.

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

S S S Sn n

g t g nT p t nT p t g nT t nT

p t g t

= − = −

=

( )g t

( )g t ( )p t

16

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

S S S Sn n

g t g nT p t nT p t g nT t nT

p t g t

= − = −

=

After Lathi & Ding, 4th ed., 2009; p. 307.

Practical Reconstruction For Non-ideal Samples – II

17

To reconstruct g(t) from non-ideal sampling, we must filter g(t) by usinga special filter known as an equalizer. In the frequency domain,

1( ) ( ) ( )S

nS

G f P f G f nfT

= −

We denote the equalizer’s transfer function as E(f). The operation weuse is to apply the filter to G(f), hence

1( ) ( ) ( ) ( ) ( ) ( )S

nS

G f E f G f E f P f G f nfT

= = =

This relationship shows that the equalizer must remove all the shiftedreplicas G(f – nfS) in the summation except for the low-pass term at n = 0.

Equalizer( )g t ( )g t

( )G f

( )G f

Signal Recovery Observations

Generally we use short pulses of durationTp to sample a function g(t). This does abetter job of signal interpolation withoutexcessive demands upon an equalizerfilter. In fact, often the equalizer can beomitted.

t

( )g t

TS

Tp

In practice it is impossible to precisely recover a band-limited signal g(t)from its samples, even if the sampling rate is higher than the Nyquist rate.

The sampling theorem rests upon the signal being strictly band-limited.All practical signals are time-limited, they therefore can’t be precisely band-limited.

18

After Lathi & Ding, 4th ed., 2009; p. 308.

Simple & Practical “Sample and Hold” Circuit

19

Basic sample and hold circuit and general waveform

Note: The RC time constantsare not shown in figure.

Also, the samples are flat-topsamples of duration Tp seconds.

VG1VG2

SamplingSwitch

DischargeSwitch

HoldingCapacitor

C

+ +

g(t) g(t)

g(t)

g(t)

t

TS = 1/fS

TS

Tp

Example of a Practical Circuit For “Sample and Hold”

20

http://www.electro-tech-online.com/attachments/samplehold-jpg.16101/

VG1VG2

C+

+

g(t)g(t)

S1

S2

++A1

A2

InputBuffer

OutputBuffer

SamplingPulse

R

Sampling capacitor C is isolated from input and output using unity-gainconnected operational amplifiers and independent input and output resistances (high input resistance and very low output resistance).

The Problem of Aliasing (or Spectral Folding)

Because signals are not band-limited, they have long tails in the frequency domain as shown in G(f). Sampling at higher rates does not eliminate spectraloverlapping of repeated spectral cycles as shown in (b).

21

Lost tail isfolded back Lost tail

0

Reconstruction filter

H(f) Sample signalspectrum

0

fs /2- fs /2 fs f

f

S

- fs

S−

( )G f

( )G f

(a)

(b)

Note the spectra cross at Frequency fS /2 = (1/2TS) Hz.

Fourier transformof waveform g(t)

s /2-s /2 s -s

After Lathi & Ding, 4th ed., 2009; p. 311.

22

1. What Can We Do to Reduce the Problem of Aliasing?

We can oversample, that is, we can sample at a rate exceeding the Nyquistrate. This is illustrated below:

B-B f

f

fS

fS

Sampled at Nyquist rate

Sampled above Nyquist rate

( )G f

( )G f

Filter

Filter

23

2. What Can We Do to Reduce the Problem of Aliasing?

We can place an anti-aliasing filter in front of the sampler.

Haa(f)

Anti-aliasing filter

Sampler( )g t ( )aag t ( )aag t

( )T t

Reconstructed filterII(f)

Reconstructed spectrum

Gaa(f)

Folded tail distortslower frequencies

Lost tail results in lossof higher frequencies

Reconstructed spectrum(no distortion of lower frequencies

Reconstructed filterII(f) Gaa(f)

Sample signalspectrum

After Lathi & Ding, 4th ed., 2009; p. 311.

Maximum Information Rate in Communications

Basic relationship in digital communications:

A maximum of 2B independent elements of information per second can be transmitted, error-free, over a noiseless channelof bandwidth B Hz.

This is related to the sampling theorem:

Remember the sampling theorem states that a low-pass signal g(t) ofbandwidth B Hz (i.e., band-limited) can be recovered from uniform samples taken at the rate of 2B samples per second.

The sampling theorem is important in signal analysis, digital signalprocessing and transmission because it allows us to replace an analogsignal with a discrete sequence of numbers (i.e., a digital signal).

24

Digital Multiplexing & Demultiplexing – TDM

25

TimeDivisionMultiplexed(TDM) Output

Time sharing a transmission medium.

MUX DEMUX

Example: Time Division Multiplexing of Two Signals

26

1( )g t

2( )g t

t

After Lathi & Ding, 4th ed., 2009; p. 319.

TDM Can Be Realized Using NAND Gates

http://www.electronics-tutorials.ws/combination/comb_2.html

27

Note: 4PST means four pole, single throw, switch.

Fou

r in

pu

tsSe

lect

Lin

es

28

Example of TDM in Operation

https://rotechproject.wordpress.com/2014/04/13/time-division-multiplexing/

29

1. Digital is more robust than analog to noise and interference†

2. Digital is more viable to using regenerative repeaters

3. Digital hardware more flexible by using microprocessors and VLSI

4. Can be coded to yield extremely low error rates with error correction

5. Easier to multiplex several digital signals than analog signals

6. Digital is more efficient in trading off SNR for bandwidth

7. Digital signals are easily encrypted for security purposes

8. Digital signal storage is easier, cheaper and more efficient

9. Reproduction of digital data is more reliable without deterioration

10. Cost is coming down in digital systems faster than in analog systems and DSP algorithms are growing in power and flexibility

† Analog signals vary continuously and their value is affected by all levels of noise.

Advantages of Digital Over Analog For Communications

30

31

Definition of Fourier Transform

1

. .The Fourier transform ( , spectrum) of is ( ):

( ) ( ) ( )

1( ) ( ) ( )

2

Therefore, ( ) ( ) is a Fourier Transform pair

j t

j t

i e f(t) F

F F f t f t e dt

f t F F F e d

f t F

−

−

−

−

= =

= =

Agbo & Sadiku;Section 2.7;

pp. 40-41

Note: Remember = 2 f

-1

32

Sampling Theorem

Sampling Theorem:

A physical waveform may be represented over the interval - < t < + by

where

and fS is the sampling rate parameter ( fS > 0). If g(t) is bandlimited to

B Hz, and fS is greater than or equal to 2B Hz, then the equation becomesthe sampling function representation with

For fS > 2B Hz, the orthogonal series coefficients are the values of thewaveform when sampled every 1/fS seconds.

( )

( )

=−

− = −

sin (

( )(

S S

nn S S

f t n fg t a

f t n f

( )

( )

−

− = −

sin (

( )(

S S

n S

S S

f t n fa f g t dt

f t n f

=

n

S

na g

f

Proof in Leon W. Couch, II, 8 edition, Section 2-7, pp. 91-93.

33

https://slideplayer.com/slide/5274806/

Uniform Linear Quantization

34

1-to-8 FET Multiplexer/Demultiplexer

http://www.ti.com/lit/ds/symlink/sn74cbt3251.pdf

https://electronics.stackexchange.com/questions/47279/differences-between-a-fet-multiplexer-and-regular-digital-multiplexer

35

Aliasing in Sampling

Aliasing 1

2ST

W

1

2ST

W

https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Book%3A_Electrical_Engineering_(Johnson)/5%3A_Digital_Signal_Processing/5.03%3A_The_Sampling_Theorem

( )S f

36https://slideplayer.com/slide/5261494/