Lecture2 Noise Freq (1)

8/18/2019 Lecture2 Noise Freq (1)

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Biomedical SignalsSignals and Images in Medicine

Lecture 2

Dr Nabeel Anwar


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Noise Removal: Time Domain

Techniques

1. Synchronized Averaging (covered in lecture 1)

2. Moving Average Filters (today’s topic)

3. Derivative based operations to remove low frequency

artifacts (today’s topic)

4. Introduction to frequency domain


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Moving Average Filters

Problem:

Propose a time-domain technique to remove random noise

given only one realization of the signal or event of interest.


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Moving Average (MA) Filter

Ensemble of several realizations not available —

• synchronized averaging is not possible.

• Consider temporal averaging for noise removal

• Temporal window of samples moved to obtain output at

various points of time: moving-window averaging or moving-

average (MA) filter.

• Average∼weighted combination of samples


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Rangaraj’s book 3.3.2

x and y: input and output of filter.

bk

: filter coefficients or tap weights.

N: order of filter.



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Basic Concept OF Z-Transform

Z-Transform is a digital operation

When identified with a digital data

sequence, such as x (n) , z−n represents an interval shift of n samples, or an

associated time shift of nTs seconds.

Every data sample in the sequence x (n) is associated with a unique power

of z, and this power of z defines a sample’s position in the sequence.

For example, the time shifting characteristic of the Z-transform can be

used to define a unit delay process, z−1. For such a process, the output is

the same as the input, but shifted (or delayed) by one data sample



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Delay of one Sample

Consider the system y[n] = x[n-1], i.e., the one-sample-delay system.

The z-transform system function is

1 z z H

Z -1


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Delay of k Samples

k z z H

Similarly, the system y[n] = x[n-k ], i.e., the k-sample-

delay system, is the z-transform of the impulse

response x[n - k ].

Z -k


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Signal-flow diagram of a moving-average filter of order N. Each block with the symbol

z−1 represents a delay of one sample, and serves as a memory unit for the

corresponding signal sample value.


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Example: Running Average Algorithm

Block Diagram Transfer Function

4

23321

4

1

4

1

z

z z z z X

z z z z X z Y

4

23

4

1

z

z z z z

X

Y

4

321

k k k k k

x x x x y

Note: Each [Z -1 ] block can be thought of as a memory cell,

storing the previously applied value.

(Non-Recursive)

Z Transform


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• Moving Average can be symmetrical or non-symmterical.

• e.g. Y[80] = ( x[80] + x[79]+x[78] +x[77]) / 4 : non-

symmterical.

Y[80] = ( x[79] + x[80]+x[81] ) / 3 : symmterical

A potential problem is look-ahead



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See Fig 15-1 in DSP Guide


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• Examples of moving average

Moving average term is used for any process that uses a moving

set of multiplier weights. That means we have finite number of

weights. It gives the concept that

An MA filter is a finite impulse response (FIR) filter:

• Output depends only on the present input sample and a few

past input samples.



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MA: Application Example

ECG signal with 1000 Hz noise

8 points moving average filter


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Moving average is Recursive

A tremendous advantage of the moving average filter is that it can be

implemented with an algorithm that is very fast.


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Assignment 2

• Apply an 8 point moving average filter on the raw ECG data

already provided.

• Apply a 32 point filter on the raw ECG data.

• What is the difference? Does 32 point filter preserved

information?


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Frequency Analysis


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Next few slides are taken from book “Advanced Engineering Mathematics by Erwin Kreyszig” John Wiley & Sons,

Inc. Consult the book for further info.

Pages 478-479b


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A d v a n c e d E n g i n e e r i n g

M a t h e m a t i c s b y E r w

i n K r e y s z i g

C o p y r i g h t

2 0 0 7 J o

h n W i l e y &

S o n s ,

I n c .

A l l r i g h t s

r e s e r v e d .


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A d v a n c e d E n g i n e e r i n g

M a t h e m a t i c s b y E r w

i n K r e y s z i g

C o p y r i g h t

2 0 0 7 J o

h n W i l e y &

S o n s ,

I n c .

A l l r i g h t s

r e s e r v e d .Pages 480-482c


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Fourier Transform

• Before we start, lets discuss a little about Fourier Transform.

• The basic idea is: A signal can be decomposed into periodic

cosine and sine waves with different frequencies.

Fourier Analysis has four categories (Slide 30-31: taken from

Smith’s book)


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Frequency Spectra

• example : g (t ) = sin(2π f t ) + (1/3)sin(2π (3f ) t )

= +


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=

Frequency Spectra


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Example: Music

• We think of music in terms of frequencies at different

magnitudes

Time information is LOST


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Fourier Transforms

All of us has basic concept of functions.

In signal processing, most functions fall into two categories:

1. waveforms, images, or other data;

2. and entities that operate on waveforms, images, or other data

The later group can be further divided into functionsi. that modify the data,

ii. and functions used to analyze or probe the data.

For example, the basic filters use functions (the filter coefficients) thatmodify the spectral content of a waveform.

The Fourier Transform detailed uses functions (harmonically relatedsinusoids) to analyze the spectral content of a waveform.

Functions that modify data are also termed operations or transformations.


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• A transform can be thought of as a re-mapping of the original

data into a function that provides more information than the

original.

• The Fourier Transform is classic example as it converts theoriginal time data into frequency information which often

provides greater insight into the nature and/or origin of the

signal.

Fourier Transform


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• How we do that?....

Many of the transforms are achieved by comparing the signal of

interest with some sort of probing function. This comparison

takes the form of a correlation (produced by multiplication) that

is averaged (or integrated) over the duration of the waveform,

or some portion of the waveform:

Fourier Transform


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Fourier Transform

It is a technique for examining signals in the frequency domain.

Our immediate goal is to represent a given function as a

convergent series in the elementary trigonometric functions

(already studied them few slides before)

f(x) F( ) Fourier

Transform


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T

nt b

T

nt a

at f

n

n

n

n

2sin

2cos

2

)(

11

0

DC Part Even Part Odd Part

T is a period of all the above signals

Fourier Analysis

with the first term

½ a0 representing the direct current (DC) component;

the remaining sine and cosine waves weighted by the an and bn coefficients

represent the alternating current (AC) components of the signal. It is very important to

find correct amplitude of ao , an and bn


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ao can be calculated by

bn all coefficients are zero except bn

General formula for an


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Fourier Analysis (in words)

The above expression has a real part of cosine of frequency f, and an imaginary part of sine of

frequency f. So what we are actually doing is, multiplying the original signal with a complex

expression which has sines and cosines of frequency f. Then we integrate this product (In other

words, we add all the points in this product). If the result of this integration (which is nothing but

some sort of infinite summation) is a large value, then we say that : the signal x(t), has a dominant

spectral component at frequency "f". This means that, a major portion of this signal is composed of

frequency f. If the integration result is a small value, than this means that the signal does not havea major frequency component of f in it. If this integration result is zero, then the signal does not

contain the frequency "f" at all.

How this integration works: The signal is multiplied with the sinusoidal term of frequency "f". If the

signal has a high amplitude component of frequency "f", then that component and the sinusoidal

term will coincide, and the product of them will give a (relatively) large value. This shows that, thesignal "x", has a major frequency component of "f".

However, if the signal does not have a frequency component of "f", the product will yield zero,

which shows that, the signal does not have a frequency component of "f". If the frequency "f", is

not a major component of the signal "x(t)", then the product will give a (relatively) small value. This

shows that, the frequency component "f" in the signal "x", has a small amplitude, in other words, it

is not a major component of "x“.


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Fourier: Examples

• The typical syntax for computing the FFT of a signal is FFT(x,N)

where x is the signal, x[n], you wish to transform, and N is the

number of points in the FFT. N must be at least as large as the

number of samples in x[n]. Matlab Example: To demonstrate

the effect of changing the value of N,

• i.e 64, 128,256, 512

• Matlab example


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Fourier: Examples

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

N = 64

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

N = 128

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

N = 256

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

N = 16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

N = 32


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Fourier: Examples

In the previous slide

• Upon examining the plot one can see that each of the transformsadheres to the same shape, differing only in the number of FFTsamples used to approximate that shape.

• But once the number of FFT points are very small the shape isdifferent (N=16 and N=32).

•When the FFT is computed with an N larger than the number ofsamples in x[n], it fills in the samples after x[n] with zeros. Forexample if x[n] is 30 samples long, and the length FFT is 256. WhenMatlab computes the FFT, it automatically fills the spaces from n=30 to n = 255 with zeros. This is called zero padding.


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1. The FFT does not directly give you the spectrum of a signal. As we

have seen with the last two examples, the FFT can vary

dramatically depending on the number of points (N) of the FFT,

and the number of periods of the signal that are represented.

2. The FFT contains information between 0 and fs, however, we

know that the sampling frequency must be at least twice the

highest frequency component. Therefore, the signal's spectrum

should be entirly below fs/2 , the Nyquist frequency.

3. Also real signal should have a transform magnitude that is

symmetrical for positive and negative frequencies. So instead of

having a spectrum that goes from 0 to fs, it would be more

appropriate to show the spectrum from –fs/2 to fs/2 .

Fourier: Examples


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Fourier: Examples

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14

16

frequency / f s


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A bit More on FFT

• It is clear that fft gives you a two sided spectrum. The basic

computations for analyzing signals include converting from a

two-sided power spectrum to a single-sided power spectrum.

Converting from a Two-Sided Power Spectrum to a Single-Sided Power Spectrum

In a two-sided spectrum, half the energy is displayed at the

positive frequency, and half the energy is displayed at the

negative frequency. Therefore, to convert from a two-sidedspectrum to a single-sided spectrum, discard the second

half of the array and multiply every point except for DC by two.


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DC component

remains same

Others are multiplied

by 2.


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FFT: Spectrum Type and Scaling

• FFT returns complex-valued amplitudes

• Real part represents cosine components,

• imaginary part represents sine components (90° phase

difference)

•Can be converted to magnitude and phase

• Squared magnitude represents signal power


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Limitations of Fourier Analysis

1. Cannot not provide simultaneous time and frequency

localization.

2. Not useful for analyzing time-variant, non-stationary signals.

3. Not efficient for representing discontinuities or sharp corners

(i.e., requires a large number of Fourier components to represent

discontinuities).


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)502cos(

)252cos(

)52cos()(4

t

t

t t f

Provides excellent

localization in the

frequency domain

but poor localization

in the time domain.



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1. Cannot not provide simultaneous time and frequency

localization.

2. In its current form Not useful for analyzing time-variant, non-stationary signals.

3. Not appropriate for representing discontinuities or sharp corners

(i.e., requires a large number of Fourier components to representdiscontinuities).


St ti t ti


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Stationary vs non-stationary

signals

• Stationary signals: time-

invariant spectra

• Non-stationary signals:

time-varying spectra

S i i


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Stationary signal:

Three frequency

components,

present at all

times!


signals

St ti t ti


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Non-stationary signal:


signals

The reason of the noise like

thing in between peaks show

that, those frequencies also

exist in the signal. But the

reason they have a small

amplitude, is because, theyare not major spectral

components of the given

signal, and the reason we see

those, is because of the

sudden change between the

frequencies.


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What's the solution to tackle the problems of non-stationary signal?

We Need a local analysis scheme for a time-frequency representation.

Windowed F.T. or Short Time F.T. (STFT)

Segmenting the signal into narrow time intervals (i.e., narrow enough to

be considered stationary).

Take the Fourier transform of each segment.

Short Time Fourier Transform


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Steps :

1. Choose a window function of finite length

2. Place the window on top of the signal at t=0

3. Truncate the signal using this window

4. Compute the FT of the truncated signal, save results.

5. Incrementally slide the window to the right

6. Go to step 3, until window reaches the end of the signal


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Each FT provides the spectral information of a separate time-

slice of the signal, providing simultaneous time and frequency

information


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2( , ) ( ) ( )u j ut f t

STFT t u f t W t t e dt

STFT of f(t):

computed for each

window centered at t=t’

Time

parameter

Frequency

parameter Signal to

be analyzed

Windowing

function

centered at t=t’

1D 2D



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Choosing Window W(t)

• What shape should it have?

• Rectangular, Gaussian, Elliptic…?

• How wide should it be?

• Window should be narrow enough to make sure that the portionof the signal falling within the window is stationary.

• Very narrow windows do not offer good localization in the

frequency domain.


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STFT Window Size

W(t) infinitely long: STFT turns into FT, providing excellent frequency localization, but no time

information.

W(t) infinitely short: gives the time signal

back, with a phase factor, providing excellent time localization butno frequency information.

1)( t W

)()( t t W

2( , ) ( ) ( ) ( )u j ut jut f t

STFT t u f t t t e dt f t e

2( , ) ( ) ( )u j ut f t

STFT t u f t W t t e dt


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STFT Window Size (cont’d)

• Wide window good frequency resolution, poor time

resolution.

• Narrow window good time resolution, poor frequencyresolution.

• In next lectures: We will study different windowing

types/shapes.


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Example

different size windows

(four frequencies, non-stationary)

Use this link for an Excellent work of Polikar on STFT and wavelets

http://users.rowan.edu/~polikar/WAVELETS/WTpart2.html


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Example


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Example


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Consider following scenario…Two sin waves have frequencies

of 100 Hz and 110 Hz, and the sampling rate is 1 kHz. Apply a

FFT with window length 64 samples and 1024 samples.

Solution: We experiment with two time windows of lengthN1=1024 with a theoretical frequency resolution of

∆f=1000/1024=0.98 Hz,

and N2=64 with a theoretical frequency resolution

∆f=1000/64=15.7 Hz.

Example: Effect of FFT window


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Heisenberg (or Uncertainty)


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Principle

4

1 f t

Time resolution: How well

two spikes in time can be

separated from each other in

the transform domain.

Frequency resolution: How

well two spectral components

can be separated from each

other in the transform domain

!t and f cannot be made arbitrarily small



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Principle

• One cannot know the exact time-frequency

representation of a signal.

• We cannot precisely know at what time instance afrequency component is located.

• We can only know what interval of frequencies are

present in which time intervals.

The Effect of Finite Length Data


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The Effect of Finite Length Data

(Windowing)

We already know that In practical situation we have either with

a short length signal, or with a long signal. Having a short

segment of N samples of a signal or taking a slice of N samples

from a signal is equivalent to multiplying the signal by a window(We have considered Gaussian shape in example) of N samples.

Multiplying two signals in time is equivalent to the convolution

of their frequency spectra. Thus the spectrum of a short

segment of a signal is convolved with the spectrum of a

rectangular pulse, the result of this convolution is somespreading of the signal energy.


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Assignment

• Take the FFT of complete single trial signal. Already provided

to you.

• Remember that FFT has both Real and Imagery Parts. So take

real part only, you can also use abs function.

• Length of FFT window should also be the part of your concept.

• Vary the fft length and look into its influence on the signal.

Fs = 150; % Sampling frequency


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Fs = 150; % Sampling frequency

t = 0:1/Fs:1; % Time vector of 1 second

f = 5; % Create a sine wave of f Hz.

x = sin(2* pi*t*f);

% x = cos(2* pi*t*f);

%x = square(2* pi*t*f);

nfft = 1024; % Length of FFT

% Take fft, padding with zeros so that length(X)

X = fft(x,nfft);

% second 0 0.2 0.4 0.6 0.8 1

%Power Spectrum of a Sine Wave FFT is symmetric, throw away half

X = X(1:nfft/2);

% Take the magnitude of fft of x

mx = abs(X);% Frequency vector

f = (0:nfft/2-1)*Fs/nfft;

% Generate the plot, title and labels.

subplot (2,1,1)

plot(t,x);

title('Sine Wave Signal');

xlabel('Time (s)');

ylabel('Amplitude');

subplot(2,1,2)

plot(f,mx);

title('Power Spectrum of a Sine Wave');

xlabel('Frequency (Hz)');

ylabel('Power');


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Spectrum Types

• Magnitude: Amplitude Spectrum

• Squared magnitude: Power Spectrum

• Squared magnitude per unit bandwidth: Power Spectral

Density

• Squared magnitude × block time length: Energy

Spectrum

• Squared magnitude × block length per unit bandwidth:

Energy Spectral Density


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Spectrum Types: When to use

• Periodic signals (discrete frequencies): Amplitude or PowerSpectrum

• Broadband random signals: Power Spectral Density

• Transient Signals: Energy Spectral Density


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Averaging in Spectrum

• Combine multiple time blocks together to form one spectralestimate

• Random data: higher average number, better estimate of

random characteristics


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Flow diagram

Lecture2 Noise Freq (1)

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