Sampling Continuous-Time Signals - CppSim

M.H. Perrott © 2007 Sampling Continuous-Time Signals, Slide 1

SamplingContinuous-Time

Signals• Impulse train and its Fourier Transform • Impulse samples versus discrete-time sequences• Aliasing and the Sampling Theorem• Anti-alias filtering• Comparison of FT, DTFT, Fourier Series

Copyright © 2007 by M.H. PerrottAll rights reserved.


The Need for Sampling

• The boundary between analog and digital– Real world is filled with continuous-time signals– Computers (i.e. Matlab) operate on sequences

• Crossing the analog-to-digital boundary requires sampling of the continuous-time signals

• Key questions– How do we analyze the sampling process?– What can go wrong?

x[n]xc(t)

t

xc(t)

n

x[n]

1

A-to-D

Converter

Real World(USRP Board)

Matlab


An Analytical Model for Sampling

• Two step process– Sample continuous-time signal every T seconds

• Model as multiplication of signal with impulse train– Create sequence from amplitude of scaled impulses

• Model as rescaling of time axis (T goes to 1)• Notation: replace impulses with stem symbols

p(t)

xp(t) x[n]xc(t) Impulse Train

to Sequence

t

p(t)

T

1

t

T

t

xc(t) xp(t)

n

x[n]

1

Can we model this in the frequency domain?


t

p(t)

T

1

P(f)

f0

T1

T2

T-1

T-2

T1

Fourier Transform of Impulse Train

• Impulse train in time corresponds to impulse train in frequency– Spacing in time of T seconds corresponds to spacing in

frequency of 1/T Hz– Scale factor of 1/T for impulses in frequency domain– Note: this is painful to derive, so we won’t …

• The above transform pair allows us to see the following with pictures– Sampling operation in frequency domain– Intuitive comparison of FT, DTFT, and Fourier Series


t

p(t)

T

1

P(f)

f0

T1

T2

T-1

T-2

T1

t

T

t

xc(t) xp(t)

Xp(f)

f0

T1

T2

T-1

T-2

TA

Xc(f)

f

A

Frequency Domain View of Sampling

• Recall that multiplication in time corresponds to convolution in frequency

• We see that sampling in time leads to a periodicFourier Transform with period 1/T


t

T

xp(t)

n

x[n]

1

Xp(f)

f0

T1

T2

T-1

T-2

TA

X(ej2πλ)

λ

0 1 2-1-2

TA

Frequency Domain View of Output Sequence

• Scaling in time leads to scaling in frequency– Compression/expansion in time leads to expansion/

compression in frequency• Conversion to sequence amounts to T going to 1

– Resulting Fourier Transform is now periodic with period 1– Note that we are now essentially dealing with the DTFT


p(t)


to Sequence

t

p(t)

T

1

t

T

t

xc(t) xp(t)

n

x[n]

1

Xp(f)

f0

T1

T2

T-1

T-2

TA

X(ej2πλ)

λ

0 1 2-1-2

TA

Xc(f)

f0

A

Summary of Sampling Process

• Sampling leads to periodicity in frequency domain

We need to avoid overlap of replicated signals in frequency domain (i.e., aliasing)


t

p(t)

T

1

P(f)

f0

T1

T2

T-1

T-2

T1

t

T

t

xc(t) xp(t)

Xp(f)

f0

T1

T2

T-1

T-2

TA

Xc(f)

f

A

fbw-fbw

fbw-fbw- fbw

T1- + fbw

T1

• Overlap in frequency domain (i.e., aliasing) is avoided if:

The Sampling Theorem

• We refer to the minimum 1/T that avoids aliasing as the Nyquist sampling frequency


p(t)

xp(t)

T

Impulse Train

to Sequence

x[n]

T 1

xc(t)

tt n

t

1/T0-1/Tf

10-1λ

1/T0-1/Tf

Example: Sample a Sine Wave

• Time domain: resulting sequence maintains the same period as the input continuous-time signal

• Frequency domain: no aliasing

Xc(f) Xp(f) X(ej2πλ)

Sample rate is well above Nyquist rate


tt n

p(t)

xp(t)

T

Impulse Train

to Sequence

x[n]

T 1

xc(t)

t

1/T0-1/Tf

10-1λ

1/T0-1/Tf

Increase Input Frequency Further …

• Time domain: resulting sequence still maintains the same period as the input continuous-time signal

• Frequency domain: no aliasing

Sample rate is at Nyquist rate



tt n

p(t)

xp(t)

T

Impulse Train

to Sequence

x[n]

T 1

xc(t)

t

1/T0-1/Tf

10-1λ

1/T0-1/Tf


• Time domain: resulting sequence now appears as a DC signal!

• Frequency domain: aliasing to DC

Sample rate is at half the Nyquist rate



tt n

p(t)

xp(t)

T

Impulse Train

to Sequence

x[n]

T 1

xc(t)

t

1/T0-1/Tf

10-1λ

1/T0-1/Tf


• Time domain: resulting sequence is now a sine wave with a different period than the input

• Frequency domain: aliasing to lower frequency

Sample rate is well below the Nyquist rate



p(t)


to Sequence

t

p(t)

T

1

Xp(f)

f0

T1

T2

T-1

T-2

TA

X(ej2πλ)

λ

0 1 2-1-2

TA

Xc(f)

f0

A

T1

T-1

Undesired

Signal or

Noise

The Issue of High Frequency Noise

• We typically set the sample rate to be large enough to accommodate full bandwidth of signal

• Real systems often introduce noise or other interfering signals at higher frequencies– Sampling causes this noise to alias into the desired

signal band


xc(t)H(f)

p(t)

xp(t) x[n]Impulse Train

to Sequence

t

p(t)

T

1

Xp(f)

f0

T1

T2

T-1

T-2

X(ej2πλ)

λ

0 1 2-1-2

Xc(f)

f0

T1

T-1

xc(t)

Anti-aliasLowpass

H(j2πf)

Anti-Alias Filtering

• Practical A-to-D converters include a continuous-time filter before the sampling operation– Designed to filter out all noise and interfering signals

above 1/(2T) in frequency– Prevents aliasing


Using the Impulse Train to Compare the FT, DTFT, and Fourier Series


p(t)


to Sequence

t

xc(t)

n

x[n]

1

X(ej2πλ)

λ

0 1 2-1-2

Xc(f)

f0

Relationship Between FT and DTFT

FT

Continuous, Non-Periodic

Non-Periodic, Continuous

Discrete, Non-Periodic

Periodic, Continuous

DTFT

Time:

Freq:


0T1

T2

T-1

T-2

T-3

T3

T4

T-4

T5

T-5

Xp(f)

f

Tp

1Tp

-10

Tp

2Tp

-2

0T1

T2

T-1

T-2

T-3

T3

T4

T-4

T5

T-5

P(f)

f f

X(f)

t t

x(t)xp(t)

t

Tp

p(t)

0 T 2T-T-2TTp

T

Relationship Between FT and Fourier Series

FT

Continuous, Non-Periodic

Non-Periodic, Continuous

Continuous, Periodic

Non-Periodic, Discrete

Fourier Series

Time:

Freq:


Summary• The impulse train and its Fourier Transform form a

very powerful analysis tool using pictures– Sampling, comparison of FT, DTFT, Fourier Series

• Sampling analysis:– Time domain: multiplication by an impulse train followed by

re-scaling of time axis (and conversion to stem symbols)– Frequency domain: convolution by an impulse train followed

by re-scaling of frequency axis

• Prevention of aliasing– Sample faster than Nyquist sample rate of signal bandwidth– Use anti-alias filter to cut out high frequency noise

• Up next: downsampling, upsampling, reconstruction

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Sampling Continuous-Time Signals - CppSim