Fundamentals of Digital Signal Processing

Lecture 8 Sampling Theorem

Fundamentals of Digital Signal ProcessingSpring, 2012

Wei-Ta Chu2012/3/20

1 DSP, CSIE, CCU

The Concept of Aliasing� Aliasing (化名): two names for the same person, or

thing

� Consider and

Aliasing is solely due to the fact that trigonometric

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� Aliasing is solely due to the fact that trigonometric functions are periodic with period

� These continuous cosine signals are equal at integervalues n

Sampled with Ts = 1

The Concept of Aliasing� The frequency of x2[n] is , while the

frequency of x1[n] is . When speaking about the frequencies, we say that is an alias of

� E.g. Show that is an alias of

� The following formula holds for the frequency aliases:

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� The following formula holds for the frequency aliases:

� Where is the smallest of all the aliases, it’s sometimes called the principal alias.

The Concept of Aliasing� Note that , so we can generate

another alias for x1[n] as follows:

� A general form for all the alias frequencies of this type

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� These aliases of a negative frequency are called folded aliases

The Concept of Aliasing� Extra relation between folded aliases and the principal

alias

folded aliases

principal aliases

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� Note that the algebra sign of the phase angles of the folded aliases must be opposite to the sign of the phase angle of the principal alias

Summary� We can write the following general formulas for all

aliases of a sinusoid with frequency

� Because the following signals are equal for all n

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Spectrum of a Discrete-Time Signal� Drawing the spectrum representation of the principal

alias along with several more of the other aliases.

� Spectrum of discrete-time signal跟spectrum of continuous-time signal的意義稍有不同� In continuous case, all the spectrum components were

added together to synthesize the continuous-time signal.

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added together to synthesize the continuous-time signal. � In discrete case, we simply need to select one spectrum

component to synthesize the discrete-time signal.

Spectrum of continuous-time signal

The Sampling Theorem� How frequently we must sample in order to retain

enough information to reconstruct the original continuous-time signal from it samples?

A continuous-time signal x(t) with frequenciesShannon Sampling Theorem

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A continuous-time signal x(t) with frequenciesno higher thanfmax can be reconstructed exactlyfrom its samplesx[n]=x(nTs), if the samples aretaken at a ratefs=1/Ts that is greater than 2fmax.

The Sampling Theorem� The minimum sampling rate of 2fmax is called the

Nyquist rate. � We can see examples of the sampling theorem in many

commercial products. � E.g. CDs use a sampling rate of 44.1 kHz for storing

music signals in a digital format. This number is slightly more than two times 20 kHz, which is the generally

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more than two times 20 kHz, which is the generally accepted upper limit for human hearing.

� Reconstruction of a sinusoid is possible if we have at least two samples per period.

� What happens when we don’t sample fast enough? � Aliasing occurs

Ideal Reconstruction� Since the sampling process of the ideal C-to-D

converter is defined by the substitution t=n/fs, we would expect the same relationship to govern the ideal D-to-C converter

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� This substitution is only true wheny(t) is a sum of sinusoids

Ideal Reconstruction� An actual D-to-A converter involves more than this

substitution, because it must also “fill in” the signal values between the sampling times, tn=nTs.

� In Section 4-4, we will see how interpolation can be used to build an A-to-D converter that approximates the behavior of the ideal C-to-D converter.

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the behavior of the ideal C-to-D converter.

� In Chapter 12, we will use Fourier transform theory to show how to build better A-to-D converters by incorporating a lowpass filter.

Ideal Reconstruction� If the ideal C-to-D converter works correctly for a

sampled cosine signal, then we can describe its operation as frequency scaling.

� For example, the discrete-time frequency of y[n] isthe continuous-time frequency of y(t) is

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� The discrete-time signal has aliases. Which discrete-time frequency will be used? � The selection is the lowest possible frequency

components (the principal aliases)� When converting from to analog frequency, the output

frequency always lies between and

Summary� The Shannon sampling theorem guarantees that if x(t)

contains no frequencies higher than fmax and if fs>2fmax, then the output signal y(t) of the ideal D-to-C converter is equal to the signal x(t)

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Spectrum View of Sampling� Suppose we start with a continuous-time sinusoid,

, whose spectrum consists of two spectrum lines at with complex amplitudes of

� The sampled discrete-time signal

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has two spectrum lines at , but it also must contain all the aliases at

Spectrum View of Sampling

� When a discrete-time sinusoid is derived by sampling, the alias frequencies all are based on the normalized value, , of the frequency of the continuous-time

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value, , of the frequency of the continuous-time signal.

� We will see what happens at different sampling rates.

Over-Sampling� Oversampling: sampling at a rate higher than twice the

highest frequency so that we will avoid the problems of aliasing and folding.

� at a sampling rate fs=500 samples/sec, we are sampling two and a half times faster than the minimum required by the sampling theorem.

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theorem.

� The input analog frequency of 100 Hz maps to , so we plot

spectrum lines at � We also draw all aliases at

Over-Sampling

Because

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� The D-to-C converter must select just one pair of spectrum lines� Always selects the lowest possible

frequency for each set of aliases (principal alias frequencies)

Over-Sampling� For the oversampling case where the original

frequency f0 is less than fs/2, the original waveform will be reconstructed exactly.

� In the example, f0=100 Hz and fs=500, so the Nyquistcondition of the sampling theorem is satisfied, and the output y(t) equals the input x(t).

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output y(t) equals the input x(t).

Aliasing Due to Under-Sampling� When fs < 2f0, the signal is under-sampled.

� For example, fs = 80 Hz and f0 = 100 Hz.

� The discrete-time frequency

� All aliases

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� Examine the D-to-C process, we use the lowest frequency spectrum lines from the discrete-time spectrum.

Aliasing Due to Under-Sampling� Another way to state this

result is to observe that the same samples would have been obtained from a 20 Hz sinusoid.

� Notice that the alias

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� Notice that the alias frequency of 20 Hz can be found by subtracting fs

from 100 Hz.

Aliasing Due to Under-Sampling� When the sampling rate and the

frequency of the sinusoid are the same.

� Samples are always taken at the same place on the waveform, so we get the equivalent of sampling a constant (DC), which is the

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a constant (DC), which is the same as a sinusoid with zero frequency.

�

� The aliases

� Principal alias frequency

Folding Due to Under-Sampling� fs=125 samples/sec

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� The one at is an alias of . This is an example of folding.

Folding Due to Under-Sampling� An additional fact about folding is that the sign of the

phase of the signal will be changed.

� If the original 100-Hz sinusoid had a phase of , then the phase of the component at would be and it follows that the phase of the aliased component at would also be .

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aliased component at would also be .

Folding Due to Under-Sampling� After reconstruction, the

phase of y(t) would be

� When we sample a 100 Hz sinusoid at a sampling rate of 125 samples/sec, we get the same samples that we

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the same samples that we would have gotten by sampling a 25 Hz sinusoid, but with opposite phase.

Maximum Reconstructed Frequency� The output frequency is always less than

� For a sampled sinusoid, the ideal D-to-C converter picks the alias frequency closet to and maps it to the output analog frequency via .

� Since the principal alias is guaranteed to lie between and , the output frequency will always lie

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and , the output frequency will always lie between and

Maximum Reconstructed Frequency� Using a linear FM chirp signal as the input, and then

listening to the reconstructed output signal.

� Suppose the instantaneous frequency of the input chirp increases according to Hz; i.e.

� After sampling, we have

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� After sampling, we have

� Once y(t) is reconstructed from x[n], what would you hear?

Maximum Reconstructed Frequency� The output cannot have a frequency higher than ,

even though the input frequency is continually increasing.

� (1) When the input frequency goes from 0 to , will increase from 0 to and the aliases will not need to be considered.

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to be considered.

(1)

Hz

Maximum Reconstructed Frequency� (2) When the input frequency increasing from

to , the corresponding frequency for x[n] increases from to , and its negative frequency companion goes from to . The principal alias of the negative frequency component goes from to . The reconstructed output

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goes from to . The reconstructed output signal will have a frequency going from to

(2)

Maximum Reconstructed Frequency� The terminology folded frequency comes from the fact

that the input and output frequencies are mirror images with respect to , and would lie on top of one another if the graph were folded about the line.

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fs = 2000 Hz