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Lecture 4:

Sampling of Continuous-Time Signals

Reza Mohammadkhani, Digital Signal Processing, 2014University of Kurdistan eng.uok.ac.ir/mohammadkhani

Signal Types2

� Analog signals: continuous in time and amplitude

� Digital signals: discrete both in time and amplitude

� Discrete-time signals: discrete in time, continuous in amplitude

� 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

Part 1: Periodic Sampling3

� Periodic sampling

� Frequency domain representation

� Reconstruction

� Discrete-Time Processing of Continuous-Time Signals

� Changing the sampling rate of discrete-time signals

Periodic (Uniform) Sampling4

� Sampling is a continuous to discrete-time conversion

� Most common sampling is periodic:

�[�] = ��(�)

where is the sampling period in second,

�� = 1/ is the sampling frequency in Hz, and

Ω� = 2π�� is sampling frequency in radian-per-second

� 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

Representation of Sampling5

Convert impulse

train to discrete-

time sequence�� � � � = �� �x

� �

� ��� �

�� �

�� �

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Can we reconstruct a sampled continuous-time

signal from its samples?

Three different continuous-time signals with the same set of sample values, that is, �[�] � ����� � ����� � �����.

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����is sampled with two different sampling rates

Part 2: Frequency domain representation8

� Periodic sampling

� Frequency domain representation

� Reconstruction

� Discrete-Time Processing of Continuous-Time Signals

� Changing the sampling rate of discrete-time signals

Sampling Effects -- Frequency Domain9

�� � � �� � � � → �� �Ω =1

2� �� �Ω ∗ � �Ω

� � � � � � − �

!"# → � �Ω =

2� � � Ω − $Ω�

%"#

�� �Ω �1 � �� � Ω − $Ω�

%"#

Frequency Domain10

FIGURE: Sampling effects in

frequency-domain.

(a) Spectrum of continuous-time

signal.

(b) Spectrum of sampling function

(c) Spectrum of sampled signal

with Ω� & 2Ω'(d) Spectrum of sampled signal

with (� ) 2('

Part 3: Reconstruction11

� Periodic sampling

� Frequency domain representation

� Reconstruction

� Discrete-Time Processing of Continuous-Time Signals

� Changing the sampling rate of discrete-time signals

Reconstruction12

Ideal Reconstruction13

*+ �

,+��Ω

Block diagram of an ideal bandlimited signal reconstruction system.

Ideal Reconstruction (2)14

Ideal bandlimited interpolation in the time domain.

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Aliasing in

non-bandlimited

signals

Reconstruction Requirements16

� A signal �� � can be reconstructed from its

samples, if

� ��(� is bandlimited,

� the sampling frequency Ω� is large enough to avoid

aliasing, and

� Sampling period/frequency is known

� Nyquist Rate:

� Ω� ≥ 2Ω'

� What about bandpass signals?!

Part 4:

Discrete-Time Processing of Continuous-Time Signals17

� Periodic sampling

� Frequency domain representation

� Reconstruction

� Discrete-Time Processing of Continuous-Time Signals

� Changing the sampling rate of discrete-time signals

Discrete-Time Processing of CT Signals18

� Overall system is equivalent to a continuous-time system� Input and output is continuous-time

� The continuous-time system depends on� Discrete-time system

� Sampling rate

� We’re interested in the equivalent frequency response � First step is the relation between xc(t) and x[n]

� Next between y[n] and x[n]

� Finally between yr(t) and y[n]

LTI Discrete-Time System19

� Input:

� � � � �� � � ./0 10"23 ��3∑ �� �Ω � � �5

3 $ %"# � � � 6

3

� Output of LTI system

� 7 � � * � ∗ � � → 8 ./0 � , ./0 � ./0� Output DT to CT

� 8+ �Ω � 9:; �Ω 8 ./23

� 8+ �Ω � 9:; �Ω , ./23 � ./23

� � 9:; �Ω , ./23 �3 ∑ �� �Ω � � �5

3 $ %"#

� � < Ω < �

8+ �Ω � ,=>> �Ω �� �Ω

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Part 5:

Changing the sampling rate of discrete-time signals21

� Periodic sampling

� Frequency domain representation

� Reconstruction

� Discrete-Time Processing of Continuous-Time Signals

� Changing the sampling rate of discrete-time signals

Changing the sampling rate 22

� A continuous-time signal �� � represented by a

discrete-time signal as follows:

� � � �� �� How can we change the sampling rate to have a

new discrete-time signal of the form:�T � � �� �T

where T ≠ ?

� ./0 V0"23 � 1 � �� � W

� � 2� $

%"#

�T ./0 V0"23X � 1T � �� � W

T � � 2�T $

%"#

Downsampling23

� Downsampling

� �Y � � � �Z � �� �Z

Downsampling24

� To avoid aliasing in downsampling by a factor of M

requires that:

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Downsampling –

Frequency Domain

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Downsampling

with aliasing

Downsampling factor M is

too large, therefore we have

aliasing!

A General Downsampling System27

� This system is also called decimator, and the

process is called decimation.

Increasing sampling rate = Upsampling

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Expander

Frequency domain29

� Fourier transform of the output of expander

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Example

System: interpolator

Process: interpolation

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For some other cases, very simple interpolation processes are

adequate. For example:

- Linear interpolation

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Changing the sampling rate by a non-integer factor

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References35

� A. V. Oppenheim and R. W. Schafer, Discrete-Time

Signal Processing, 3rd Edition, Prentice Hall, 2009.

� D. Manolakis and V. Ingle, Applied Digital Signal

Processing, Cambridge University Press, 2011.

� Miki Lustig, EE123 Digital Signal Processing, Lecture

notes, Electrical Engineering and Computer Science,

UC Berkeley, CA, 2012. Available at:http://inst.eecs.berkeley.edu/~ee123/fa12/