Sampling of Continuous-Time Signals

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Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall Inc.

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Signal Types• Analog signals: continuous in time and amplitude

– Example: voltage, current, temperature,…• Digital signals: discrete both in time and amplitude

– Example: attendance of this class, digitizes analog signals,…• Discrete-time signal: discrete in time, continuous in amplitude

– Example:hourly change of temperature in Austin• Theory for digital signals would be too complicated

– Requires inclusion of nonlinearities into theory• Theory is based on discrete-time continuous-amplitude signals

– Most convenient to develop theory– Good enough approximation to practice with some care

• In practice we mostly process digital signals on processors– Need to take into account finite precision effects

• Our text book is about the theory hence its title– Discrete-Time Signal Processing

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Periodic (Uniform) Sampling• Sampling is a continuous to discrete-time conversion

• Most common sampling is periodic

• T is the sampling period in second• fs = 1/T is the sampling frequency in Hz• Sampling frequency in radian-per-second s=2fs rad/sec• Use [.] for discrete-time and (.) for continuous time signals• This is the ideal case not the practical but close enough

– In practice it is implement with an analog-to-digital converters– We get digital signals that are quantized in amplitude and time

nnTxnx c

-3 -2 2 3 4-1 10

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Periodic Sampling• Sampling is, in general, not reversible• Given a sampled signal one could fit infinite continuous signals

through the samples

0-1

20 40 60 80 100

-0.5

0

0.5

1

• Fundamental issue in digital signal processing– If we loose information during sampling we cannot recover it

• Under certain conditions an analog signal can be sampled without loss so that it can be reconstructed perfectly

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Representation of Sampling• Mathematically convenient to represent in two stages

– Impulse train modulator– Conversion of impulse train to a sequence

Convert impulse train to discrete-time sequence

xc(t) x[n]=xc(nT)x

s(t)

-3T-2T 2T3T4T-T T0

s(t)xc(t)

t

x[n]

-3 -2 2 3 4-1 10n

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Continuous-Time Fourier Transform• Continuous-Time Fourier transform pair is defined as

• We write xc(t) as a weighted sum of complex exponentials• Remember some Fourier Transform properties

– Time Convolution (frequency domain multiplication)

– Frequency Convolution (time domain multiplication)

– Modulation (Frequency shift)

dtetxjX tjcc

dejX21tx tj

cc

)j(Y)j(X)t(y)t(x

)j(Y)j(X)t(y)t(x

otj jXe)t(x o

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Frequency Domain Representation of Sampling• Modulate (multiply) continuous-time signal with pulse train:

• Let’s take the Fourier Transform of xs(t) and s(t)

• Fourier transform of pulse train is again a pulse train• Note that multiplication in time is convolution in frequency• We represent frequency with = 2f hence s = 2fs

n

nTt)t(s

n

ccs nTttxtstxtx

k

skT2jS

jSjX2

1jX cs

k

scs kjXT1jX

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Frequency Domain Representation of Sampling• Convolution with pulse creates replicas at pulse location:

• This tells us that the impulse train modulator– Creates images of the Fourier transform of the input signal – Images are periodic with sampling frequency– If s< N sampling maybe irreversible due to aliasing of images

k

scs kjXT1jX

jXc

jXs

jXs

N-N

N-N s 2s 3s -2s s 3s

N-N s 2s 3s -2s s 3s

s<2N

s>2N

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Nyquist Sampling Theorem• Let xc(t) be a bandlimited signal with

• Then xc(t) is uniquely determined by its samples x[n]= xc(nT) if

N is generally known as the Nyquist Frequency• The minimum sampling rate that must be exceeded is known

as the Nyquist Rate

Nc for 0)j(X

Nss 2f2T2

jXs

jXs

N-N s 2s 3s -2s s 3s

N-N s 2s 3s -2s s 3s

s<2N

s>2N

Low pass filter

Sample Question I

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Sample Question II

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