Discrete-time Signals & Systems S Wongsa 1 Dept. of Control Systems and Instrumentation Engineering, KMUTT JAN, 2011 Overview Signals & Systems Continuous & Discrete Time Sampling Sampling in Frequency Domain Sampling Theorem 2 Aliasing & Anti-Aliasing Filter
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Discrete-time Signals & Systems
S Wongsa
11
S WongsaDept. of Control Systems and Instrumentation Engineering,
KMUTT
JAN, 2011
Overview
� Signals & Systems
� Continuous & Discrete Time
� Sampling
� Sampling in Frequency Domain
� Sampling Theorem
22
� Sampling Theorem
� Aliasing & Anti-Aliasing Filter
Lecture plan
Lecture Date Topic
1 4 & 5 Jan 11 Discrete-time signals and systems; Sampling of continuous-time signals
Discrete-time processing of continuous-time signals
Sampling Reconstruction
1414Source: 6.003 Signals & Systems, MIT, Fall 2009.
Sampling
Sampling is the process of getting a discrete signal from a continuous one.
It enables the processing of signal by digital computer.
T
)(tx )(txs
1515
• Discrete-time signal
K,2,1,0 ],[)()( ±±=== nnxnTxtxs
T
where T is a sampling time.
Sampling
We would like to sample in a way that preserves information, whichmay not seem possible because information between samples is lost.
1616Source: 6.003 Signals & Systems, MIT, Fall 2009.
How can we minimise the distortion of reconstructed signal?
Sampling
)(tx
)(txs
)(tTδ
X
1717
)()()( ttxtxTsδ=
where
∑∞
−∞=
−=n
TnTtt )()( δδ
Sampling in frequency domain
The Fourier transform of : )(txs
∑∞
−∞=
−=k
sskX
TX )(
1)( ωωω
where
Ts
πω
2= is the sampling frequency in rad/sec.
1818
T
Determine under what conditions we get:Reconstructed CT signal = Original CT signal
Goal:
Sampling in frequency domain
• If x(t) has bandwidth B and if Bs
2>ω
∑∞
−∞=
−=k
sskX
TX )(
1)( ωωω
x(t) is a bandlimited signal.
T
1919
The high frequency copies can be removed with a low-pass and then multiplying by T to undo the amplitude scaling.
T
Sampling theorem
A bandlimited signal with bandwidth B can be reconstructed completely and
exactly from its samples as long as they are taken at rate Bs
2>ω
• is called the Nyquist sampling frequency / Nyquist rate.Bs
2=ω
2020
NB: Sampling at Nyquist rate is only possible if an IDEAL lowpass filter is used. In practice we generally need to choose a sampling rate above the Nyquist rate.
What if the samples are not taken fast enough?
B-B
)(ωs
X
Aliasing Aliasing
2121
The high frequency components of x(t) will be transposed to low-frequency components, leading to a phenomenon called aliasing.
What if the signal is not bandlimited?
2222
• For non-bandlimited signal aliasing always happens regardless of value.sω
Aliasing
• What are the consequences of aliasing?
- it makes two continuous sinusoids of different frequencies indistinguishable when sampled.
2
3
2323
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3
-2
-1
0
1
Time (sec)
Am
plit
ude
Aliasing: a 52 Hz sinusoid sampled at 50 Hz.
Aliasing
• What are the consequences of aliasing?
- a distorted version of the original signal x(t).
• original music sampled at 44.1kHz (CD-quality)
Example:
• The at 4kHz downsampled version.
2424
• The at 4kHz downsampled version.
Anti-Aliasing FilterTo avoid aliasing, in practice we use a CT lowpass filter before the ADC to restrict the bandwidth of a signal to approximately satisfy the sampling theorem.
Fs = 44.1 kHz
2525Source: Prof. Mark Fowler, EECE 301 Signals & Systems, Binghamton University.
Suggested Readings
• Steven W. Smith, Chapter 3: ADC and DAC, The Scientist and Engineer's Guide to Digital
Signal Processing
2626
Review Questions
1. If we used x(t) below and sampled it at 20 kHz, how many samples would we have after 60 ms?