Biomedical Computing: Sampling and Aliasing in Discrete-Time Signals, Study notes of Signals and Systems Theory

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BME 310 Biomedical Computing -

J.Schesser

Sampling and Aliasing

Lecture

Chapter 4

BME 310 Biomedical Computing -

J.Schesser

What Is this Course All About? • To Gain an Appreciation of the

Various Types of Signals and Systems

• To Analyze The Various Types of

Systems

• To Learn the Skills and Tools needed

to Perform These Analyses.

• To Understand How Computers

Process Signals and Systems

BME 310 Biomedical Computing -

J.Schesser

Sampling

We can obtain a discrete-time signal by sampling acontinuous-time signal at equally spaced timeinstants,

t

n

nT

s

x

[

n

] =

x

nT

s

n

The individual values

x

[

n

] are called the samples of

the continuous time signal,

x

t

The fixed time interval between samples,

T

s

, is also

expressed in terms of a sampling rate

f

s

(in samples

per second) such that:

f

s

T

s

samples/sec.

BME 310 Biomedical Computing -

J.Schesser

Continuous-to-Discrete Conversion

By using a Continuous-to-Discrete (C-to-D)converter, we can take continuous-time signals andform a discrete-time signal.

There are devices called Analog-to-Digital converters(A-to-D)

The books chooses to distinguish an C-to-D converterfrom an A-to-D converter by defining a C-to-D as anideal device while

A-to-D converters are practical

devices where real world problems are evident.

• Problems in sampling the amplitudes accurately– Problems in sampling at the proper times

BME 310 Biomedical Computing -

J.Schesser

Discrete-Time Sinusoidal Signals

Since a Fourier series can be written for any continuous-timesignal, let’s concentrate on sinusoids

We define a normalized frequency for the discrete sinusoidalsignal.

is the normalized or discrete-time frequency

Since we can have different signals with the same

, then

there can be an infinite number of continuous-time signalwhich yield the same discrete-time sinusoid!

[ ]

cos(

cos(

s

s

s

s

x n

x nT

A

nT

A

n

T

f















ˆ

ˆ

BME 310 Biomedical Computing -

J.Schesser

Two Problems with Sampling

Problem 1: How many samples are enough tohave to represent a continuous time signal?

In this figure, we have a continuous-time signalsampled every .4 seconds (red samples) andevery 1 second (black samples).

(^10) -

0

2

4

6

8

10

BME 310 Biomedical Computing -

J.Schesser

161

Discrete-Time Sinusoidal Signals







4 .

) 1

)(

2. 0 ( 2 ˆ





ˆ

[ ]

cos(

)

ˆ

2

(1.2)(1)

2

.

[ ]

cos(2.

)

cos(

)

cos(0.

)

x n

A

n

x n

A

n

A

n

n

A

n



























































(^10) -

0

2

4

6

8

10

In the first case, where

f

Hz, we have:

Since a sinusoid is periodic in 2



, then for the case where

f

=1.2 Hz

BME 310 Biomedical Computing -

J.Schesser

Aliasing

This exampleillustrates that twosampled sinusoids canproduce the samediscrete-time signal.

cos [

t

]

cos [

t

]

(^10) -

0

2

4

6

8

10

When this occurs we say that that these signalsare aliases of each other.

BME 310 Biomedical Computing -

J.Schesser

164

Aliasing

Let’s look at signals of the form: cos(

l

t

Hz

c

....rad/se ,

Then, . 1

and

is

ˆ

example,

our

In

and

ˆ

or

and

ˆ

have

can

then we , )

cos(

cos(

cos(

since

and

Therefore,

integer.

an

is

and

alias,

principal

the

is

ˆ ,

and

where

cos(

cos(

. ,.. ,. l l f l f

l

l

f

f

l

f

l

f l f l f l f f l f

l

l

f

T

n

t

s

o

l

o

l

s

o

s

o

l

o

l

s o l o l s o l s o s l l

o

o

l

s

l

l s

l

l

sampled

l

BME 310 Biomedical Computing -

J.Schesser

Shannon’s Sampling Theorem

• How frequently do we need to sample?• The solution: Shannon’s Sampling Theorem:

A continuous-time signal

x

t

) with frequencies

no higher than

f

max

can be reconstructed

exactly from its samples

x

[

n

] =

x

nT

s

), if the

samples are taken a rate

f

s

T

s

that is

greater than 2

f

max

• Note that the minimum sampling rate, 2

f

max

is called the Nyquist rate.

BME 310 Biomedical Computing -

J.Schesser

Nyquist Rate

• Shannon’s theorem tell us that if we have at

least 2 samples per period of a sinusoid, wehave enough information to reconstruct thesinusoid.

• What happens if we sample at a rate which is

less than the Nyquist Rate?

– Aliasing will occur!!!!

BME 310 Biomedical Computing -

J.Schesser

Ideal Reconstruction

The sampling theorem suggests that a process existsfor reconstructing a continuous-time signal from itssamples.

If we know the sampling rate and know its spectrumthen we can reconstruct the continuous-time signal by scaling

the

principal alias

of the discrete-time signal

to the frequency of the continuous signal.

The normalized frequency will always be in the rangebetween

and be the principal alias if the

sampling rate is greater than the Nyquist rate.

BME 310 Biomedical Computing -

J.Schesser

Oversampling

When we sample at a rate which is greater than the Nyquistrate, we say we are oversampling.

If we are sampling a 100 Hz signal, the Nyquist rate is 200samples/second =>

x

t

)=cos(

t

If we sample at 2.5 times the Nyquist rate, then

f

s

samples/sec

This will yield a normalized frequency at 2

(^10) -

0

BME 310 Biomedical Computing -

J.Schesser

Oversampling

Since we are greater than the Nyquist rate, the normalizedfrequency will be <

which means it is the principal alias.

And we get back the original continuous frequency when wedo the reconstruction

f

f

s

= 0.2 (500) = 100 Hz

π

π

π

π

π

π