Phase b Frequency Estimation

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FINAL MINI PROJECT REPORT ON

NOISY SINUSOID

BY- NAGENDRA KUMAR (124102042)

MANISH SHAW (124102041)

( Group No.- 14)

UNDER THE GUIDANCE OF

DR. A.K. KARTIK


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1.A BREIF ANALYSIS OF THE PROBLEM STATEMENTThe given problem statement is to detect and identify the frequency and phase of a discrete

sinusoid which has been corrupted by white noise of different variances.

Mathematically, the signal can be represented by

x[n] = sin(n + ) + N(0 , )

Where N(0 , ) represents the Additive White Gaussian Noise (AWGN) component with fixedzero mean and variable variance. And, is phase angle lies between [0,].

The procedure followed in this work has been developed in the following steps

Frequency estimation:-

A. Signal Input.B. Frequency domain transformation ( DFT / FFT Transform).C. Frequency estimation from maximum peak.D. Band pass filtering around estimated frequency.E. Inverse transformation into time domain ( IDFT/ IFFT ).F. Smoothing the signal by LOESS method.

Phase Estimation:

G. Estimated every zero-crossing of the signal.H. Estimated phase for every zero-crossing.I. Took average of estimated phase.

2.FREQUENCY ESTIMATION2.1 INTRODUCTIONThe usual method of estimating a signal corrupted by additive noise is to pass it

through a filter that tends to suppress the noise while leaving the signal relatively unchanged

i.e. direct filtering.

Filters used for direct filtering can be either Fixed or Adaptive.

1. Fixed filters - The design of fixed filters requires a priori knowledge of both the signaland the noise, i.e. if we know the signal and noise beforehand, we can design a filter that passes

frequencies contained in the signal and rejects the frequency band occupied by the noise.


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2. Adaptive filters - Adaptive filters, on the other hand, have the ability to adjust their

impulse response to filter out the correlated signal in the input. They require little or no a

priori knowledge of the signal and noise characteristics.(If the signal is narrowband and noise

broadband, which is usually the case, or vice versa, no a priori information is needed; otherwise

they require a signal (desired response) that is correlated in some sense to the signal to be

estimated.) Moreover adaptive filters have the capability of adaptively tracking the signal under

non-stationary conditions.

During frequency estimation process, first we find out the frequency before passing signal

through filter. So we used fixed filter. And, during phase estimation process we have used

adaptive process to estimate phase if frequency was not known by transmitter and receiver.

2.2 Frequency domain transformation ( DFT Transform):

We received the signal in time domain. Generally we do not have any procedure to get

information about frequency through time domain signal. So we have to find an approach toconvert any another domain that can give the information about frequency. And most important

thing of transformation process should be that we can get the original signal from that domain by

inverse transform.

One of transformation process that can convert from time domain to frequency domain and vice-

versa is Discrete Fourier Transform/ Inverse Discrete Fourier Transform (DFT/ IDFT). So, we

have used Discrete Fourier Transform/ Inverse Discrete Fourier Transform ( DFT/ IDFT ) in our

project.

The mathematics of the above mention transform is as follows

DFT: F(k) = DFT{f(n)} = where k = 0, 1, , N-1.

Since MATLAB does not support zero indices, the values are shifted by one index value to

F(k) = DFT{f(n)} = where k = 1, , N.

( Because we did our analysis through Matlab software).

The inverse transform follows accordingly as accordingly as

IDFT: f(n) =

{F(k)} =

where n = 1, , N.

In Matlab, the function fft computes the discrete Fourier Transform of the time domain signal

and it assumes left corner as the zero.Hence, the fftshift command is used to reset the zero point

to the centre of the plot. This way, both positive and negative frequency components can be

verified and operated on. In cases where the length of the data is a power of 2, or a product of

prime factors, fast Fourier transform (FFT) algorithms are employed to compute the discrete


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Fourier transform. And the function ifft computes the inverse discrete Fourier Transform of the

frequency domain signal.

2.3 Frequency estimation from maximum peak:

After applying the discrete Fourier Transform of the received signal, we plot its absolutemagnitude graph. Graph is given below.

Figure 1: DFT of noisy sinusoid

We observed two main points from above plot.

a. Most of signal lies near about two point (i.e. nearby positive and negative component oforiginal signal).

b. Plot is symmetric about mid-point of x-axis. (i.e.

+ 1). Where N indicates the length of

data.

The generation of the frequency axis depends on the sampling rate of the input signal in the

time domain. Hence, the values from -0.5 to 0.5 are uniformly split into the number of

samples of the source signal since any signal which has been sampled higher than its nyquist

rate has its spikes lying between and which translates to the frequencies of range -0.5

and 0.5. So we translate [0,N] into the discrete frequency range by following formula.

Discrete frequency = -0.5 +

Original sinusoidal signal contains only positive and negative component of frequency but

Additive White Gaussian Noise (AWGN) contain all frequency range between [- , +] rad.

So after getting mixed original signal with additive white gaussian noise received signal

contains all frequency component between [-0.5, 0.5]. But most of signal lies near by the

original sinusoidal frequency.

PEAK IDENTIFICATION


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Now, we perform a peak identification of the fourier transform of the noisy sinusoid. The

Fine Peak Identification is performed as follows through function pid.

The increasing and decreasing trends of the curves are detected and labelled. Each change from decreasing to increasing is considered as the border between two

sub-peaks and is stored. Hence, the area between two such successive values holds a sub-peak of the plotted

frequency domain representation.

After getting the index value of maximum peak, we convert that index value in frequency by

the following formula.

Estimated sinusoidal frequency = -0.5 +

2.4 Band pass filtering around estimated frequency:

An ideal band pass filter would have a completely flat pass band (e.g. with no gain/attenuation

throughout) and would completely attenuate all frequencies outside the pass band. In practice, no

band pass filter is ideal. The filter does not attenuate all frequencies outside the desired

frequency range completely; in particular, there is a region just outside the intended pass bandwhere frequencies are attenuated, but not rejected.

Additive White Gaussian Noise (AWGN) contains sinusoids of all frequencies from - to .

Hence, it is logical to filter the noise of those frequencies which are not close to the sinusoidfrequency ( i.e. estimated frequency). This can easily be done with the help of a band pass filter.

A band pass filter is a system which allows only sinusoids whose frequencies fall within the

pass band of the filter to pass through it. An ideal band pass filter is characterized by a transfer

function which looks like below.


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Figure 2: An ideal band pass filter

2.5 DFT of noisy sinusoid after filtering

Figure 3: DFT of noisy sinusoid after filtering

From above figure, we observed that it contains the signals only nearby signal frequency and

band pass filter makes other frequency component to zero.

2.6 Inverse transformation into time domain:


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This frequency domain signal is then reset with its zero to the left end by using the MATLAB

function ifftshift and the Inverse Discrete Fourier Transform is then computed to obtain a

signal with much lesser noise.

2.7 Smooth the estimated signal:

We can minimize some percentage of error in estimated sinusoid by smooth process. We have

several methods to smooth the corrupted signal. Our signal was corrupted by White noise.

LOESS smooth gives better approximation for signal that is corrupted by white noise. So, In our

project we have used LOESS method to smooth the estimated signal.

LOESS : It function as local regression using weighted linear least squares and a 2nd

degreepolynomial model.

3.PHASE ESTIMATIONWe have several method to estimate the phase like cross-correlation method, zero-crossing

method etc. We are using zero-crossing method because in our case signal get corrupted by

Additive White Gaussian Noise (AWGN) and it contains all frequency range from 0 to . It

means original gets effected at every frequency, due to this amplitude of original discrete signal

get change at every points. It results in phase change between original signal and estimated

signal. If we have previous information about frequency (here information about frequencymeans exact frequency not estimated frequency) then we can calculate phase through the help of

zero crossing of the signal .In case 4 receiver does not signal frequency. So first we calculated

the frequency and we observed that it was not exact signal frequency. So if we compute phase

from this it leads more error in phase estimation process. So, we tried to modify the time period

for every zero-crossing. Formula to find the time-period, we have discussed later . We are

explaining the zero-crossing algorithm by the help of following figure.


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Figure 4: Shows relation between two consecutive zero-crossing in discrete signal vs. time

period in continuous signal

From above continuous sinusoid plot we observe that the time period of plot is 12.5 units. It

means 12.5 units change in horizontal direction cause of 2 radians phase cahnge.

So 1 unit horizontal change cause of phase change

radians =

radians. (Because here T =

12.5 units)

Similarly, X units horizontal change cause of phase change * X ) radians =

radians

And we can also say that

units change cause of phase change

= radians. Here is time

period of zero-crossing. In other words, we can also say that distance between any two


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consecutive zero-crossing causes of phase change radians and this property is applicable for

discrete sinusoids also.

3.1 Procedure to find out the phase:

In our problem phase is random variable with a uniform distribution in the interval [0, ] butwe mention earlier phase get change due to Additive White Gaussian Noise (we can observe

same in figure 3). So phase in estimated signal go beyond the interval [ 0, ]. So we are

consideringphase lies in the interval [ 0, 2 ].

An exhaustive search for the correct phase from 0 to 2 will be highly time consuming. Hence,

first we divide phase in two parts 0 to and to 2.

Case 1: Sinusoid starts in the positive half-cycle (positive phase) then phase lies between 0 to .

Case 2: Sinusoid starts in the negative half-cycle (negative phase) then phase lies between to

2.

Case 1: Let the first zero crossing takes place at ( can be in fraction, Procedure to find outM , we will explain after this).

Now the phase of the sinusoid from the first zero crossing can be stated as

= - * , Here T is time period for continuous sinusoid because in

discrete sinusoid more than two zero crossing can take place.

Let the second zero crossing takes place at

Phase of the sinusoid from the second zero crossing can be stated as

= 2 - * ,

Similarly we calculated up to last zero-crossing of the signal and took average of allphase. By this, we get better approximation of phase.

Case 2: We have similar process to calculate phase as case 1 except extra radian phase.

i.e. = + .

Formula to find out distance between origin and zero-crossing ( ) :

= ( Number of samples before ith zero-crossing)

+

Formula to find out time-period for every zero-crossing( ) :


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Figure 8: Input sinusoid and estimated sinusoid when receiver does not know signal frequency

and phase is non-zero.

From case 3 and case 4 we observe in case 4 estimated phases is more with respect to case 3 for

same frequency, phase and number of samples. There is only one difference that in case 3

receiver knew the sinusoid frequency where as in case 4 receiver did not know sinusoidal

frequency.

From the above estimated signal, we observed that signal cant remain exact discrete periodic

and previously mentioned during theory explanation, we need information about frequency of

signal. So in case 4 we calculated the time period for every zero-crossing (i.e. adaptive method).We used estimated frequency only for first zero-crossing and for remaining zero-crossing we

used new time period that we calculated from the zero-crossing because every two consecutive

zero-crossing cause of radians phase and radians cause half of time period in continuous

sinusoids. Through this method we get updated time period for every zero-crossing and we got

better estimation than fixed estimatedfrequency.

5. ANALYSIS OF THE OBTAINED RESULTS

The analysis of results is done on following basic aspects.

Change with increasing variance, while keeping the frequency constant. Change with increasing no. of samples Estimation of period of x[n] by examining y[n]. Average error in estimating the phase while receiver knows sinusoidal frequency.


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Average error in estimating the phase while receiver does not know sinusoidalfrequency.

Information conveyed (Entropy) in above mention four cases.5.1 INCREASING VARIANCE

Plot for Variance = 0.05

Figure 9: Transmitted signal, AWGN with N(0, 0.05), received signal and estimated signal



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Plot for Variance = 1

Figure 11: Transmitted signal, AWGN with N(0, 1), received signal and estimated signal


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From above plot, We observed that as the value of variance increases, it means SNR decreases.

So we are getting more deviation in the reconstructed sinusoidal signal. This can be attributed to

the following reasons:

The ideal band pass filter removes noise far from the frequency of the sinusoid. It does not remove the noise presents nearby the frequency of the sinusoid. It means

some noise remains in signal after passing through band pass filter also.

5.2 INCREASING NUMBER OF SAMPLES


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Estimated frequency (in rad/sec) for variance = 0.05 i.e. SNR = 10 dB

Table 1: Estimated frequency and estimated error for deferent frequency and fix variance = 0.05

Estimated frequency in(rad/sec) for variance = 0.2 i.e. SNR =3.98 dB

Table 2: Estimated frequency and estimated error for deferent frequency and fix variance = 0.2

Estimated frequency(rad/sec) for variance = 1 i.e. for SNR = -3.01 dB


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In case 512 samples are too high and 64 samples is too inaccurate, 256 can be chosenas an intermediate with reasonable accuracy and computational time.

Hence, the accuracy of estimated frequency depends on the number of samples and very

less on SNR (practically independent from SNR).

5.3 Estimation of period x[n] by examining y[n]:

By examining y[n], We are not able to determine period of x[n] more precisely in every case,

especially when SNR is negative. But, if we are taking the average distance between every two

consecutive zero crossing point of signal. Then we are able to determine period of x[n] more

precisely.

5.4 Average error in estimating the phase while receiver knows sinusoidal

frequency:

In phase estimation process, while keeping other variable constant example. frequency, numberof sample etc. we get different phase for same input phase, if we repeat experiment again and

again unlike frequency estimation process where we get estimated frequency same, if we repeat

experiment again and again while keeping other variable constant. So to get more accurate phase

we took average of five such experiment.

Cause of different estimated phase for same input phase:

To understand this, we are taking 3 samples of Additive White Gaussian Noise (AWGN) N(0,

0.2) (i.e. in our case it results in SNR=10).

Sample: A


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Sample: B

Sample: C

We observed from above three Additive White Gaussian Noise (AWGN) samples, it does not

have same amplitude at same index number. Example, at index one, sample A has amplitude

0.22, sample B has amplitude -0.1 and sample C has amplitude 0.65. Due to this receiver

receives different sinusoid sample even if we keep constant every parameter like as frequency,

phase and number of samples. Since our algorithm is based on amplitude and index value of the

signal, error in amplitude cause of error in estimated phase.

When signal frequency fo is 0.08 rad/sec


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6. M1-T1-APeriod of discrete sinusoid x[n]A discrete-time sinusoid can be expressed as

X[n] = Acos(wn + ), - < n <

Where n is an integer.

Properties:

A discrete-time sinusoid is periodic only if its frequency f is a rational number. Discrete-time sinusoids whose frequencies are separated by an integer multiplde of 2 are

identical.

So, The highest rate of oscillation in a discrete-time sinusoid is attained when w = (orw = -) or, equivalently f= 0.5 Hz ( or f= -0.5 Hz).

Period for different value of of:A. = /5

=

Where N is Time period of discrete sinusoids & a rational number.

; N=10;

B. = 3/5N=

;

C.

=

; N=2;

D. = 2N= , Here N is not a rational number, so for= 2 discrete sinusoid is not periodic.

Estimation of Time-period through Matlab using STEM() function:

A. = /5Discrete plot for given frequency:


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By above plot, we observe that signal is repeating after N=10. So, we can say that time-period

for above frequency is 10 i.e. N=10.

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Discrete plot for given frequency:

By above plot, we observed that signal complete the cycle 18 times between 0 to 600.

Time-period N= (600 * 0.1)/18=

=

Discrete plot for given frequency:


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7. Codes for phase and frequency estimation:%Codes for frequency Estimationtax_length=256;t = 1:tax_length;

tin = 1:10000;% A random frequency is generated

fo =(rand(1))/2; %It limits the frequency between [0 0.5]

disp(sprintf('Suppilied Frequency= %g', fo));

% The sinusoid is generated

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

xn = awgn(x,4.77); % AWGN is added to the sinusoid

% The frequency axis is generated

f = linspace(-0.5,0.5,length(t));

% The fourier transform of the sinusoid is computedxnfft = fft(xn);

recons=ifft(xnfft);

xnfft1 = fftshift(xnfft);

xnfft1 = abs(xnfft1);

% Peak Identification of the FFT is done and the index with the maximum

% value is found

peaks = pid(xnfft1);

m_index = mpeak(xnfft1,peaks);

%stem(m_index);

% The index is then translated to the respective value of the frequencyfreq =-0.5+ m_index/length(t);

mf = abs(freq);

disp(sprintf('Estimated Frequency= %g', mf));

%nOW, THE bandpass filtering is done to obtain the filtered sinusoid

nl=-mf-5/length(t);

nh=-mf+ 4/length(t);

pl= mf-4/length(t);

ph= mf+5/length(t);

xbp=bfilt(xn,nl,nh,pl,ph);

bpfft=fft(xbp);

bpfft1=fftshift(bpfft);bpfft1=abs(bpfft1);

subplot(4,1,1); stem(x);

xlabel('x[n],Input sinusoid,')

ylabel('Amplitude')

subplot(4,1,2); stem(xn-x);

xlabel('w[n],AWGN, mean=0')


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ylabel('Amplitude')

subplot(4,1,3); stem(xn);

xlabel('y[n],Noisy sinusoid')

ylabel('Amplitude')

subplot(4,1,4); stem(ifftshift(xbp));

xlabel('x~[n],Reconstructed Signal')

ylabel('Amplitude')

%plot of FFT for Before and after filtering

%subplot(2,1,1); stem(xnfft1);

%xlabel('DFT of Noisy sinusoid')

%subplot(2,1,2); stem(bpfft1);

%xlabel('DFT of Noisy sinusoid after filtering')

%-----------PHASE ESTIMATION--------------------if xbp(1) >= 0

check = 1;xtra_ph=0;else

check = 2;xtra_ph = pi;

end

cur_pos=1;reset_vl=0;extra_ph=0;pse_ctr=1:100;%tot_ph=1;ph_ctr=1;sum_phase=0;tp_curnt=0;fst_zc=0;sec_zc=0;tp_act=0;while (cur_pos < tax_length)

if check ==1if xbp(cur_pos)


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endcheck=2; %It check every zero cross-overreset_vl=0;sum_phase= (phase(ph_ctr)) + sum_phase;%disp(sprintf('Estimated Phase through check 1= %g, ph_ctr = %g,

timeP = %g', phase(ph_ctr), ph_ctr,(tp_curnt)));ph_ctr= ph_ctr + 1;continue;

elsereset_vl = reset_vl + 1;cur_pos=cur_pos + 1;

endendif check ==2

if xbp(cur_pos) >=0extra_phase= ((xbp(cur_pos-1)) / ((xbp(cur_pos-1) -

xbp(cur_pos)))); %extra_phase actually gives information about xtra timeperiod

if ph_ctr ==1tp_curnt = 1 / (mf * 1.02);fst_zc= (cur_pos -1) + extra_phase;phase(ph_ctr)= ((ph_ctr) * pi) - (((cur_pos -1) + extra_phase)

* ((pi * 2)/( tp_curnt))) + xtra_ph;elseif ph_ctr == 2

sec_zc = (cur_pos - 1) + extra_phase;tp_curnt = 2 * (sec_zc - fst_zc);phase(ph_ctr) = ((ph_ctr)* pi ) - (((cur_pos - 1) +

extra_phase) * ((pi * 2)/( tp_curnt))) + xtra_ph ;else

tp_curnt = (cur_pos -1) - fst_zc + extra_phase;tp_curnt = (tp_curnt /(ph_ctr - 1 )) *2;phase(ph_ctr) = ((ph_ctr)* pi ) - (((cur_pos - 1) +

extra_phase) * ((pi * 2)/( tp_curnt))) + xtra_ph;

endcheck=1; %It check every zero cross-overreset_vl=0;sum_phase= (phase(ph_ctr)) + sum_phase;

% disp(sprintf('Estimated Phase through check 2= %g, ph_ctr = %g',phase(ph_ctr), ph_ctr));

ph_ctr= ph_ctr + 1;continue;

elsereset_vl = reset_vl + 1;cur_pos= cur_pos + 1;

endend

endtotal_phase= sum_phase / (ph_ctr - 1);disp(sprintf('input Phase = %g, Final Estimated Phase = %g, SNR= 10',phi,total_phase));%------Phase Estimation process end ---------------


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%Checks for increasing and decreasing trends of the plot to decide the peak

%positions

function [ peak2 ] = pid( plot1 )

[length breadth] = size(plot1);

k=1;

for i = 2:breadth

if(plot1(i) > plot1(i-1))peak1(k) = plot1(i);

k = k+1;

else

peak1(k) = 0;

k = k+1;

end

end

peak2(1) = 1;

[L B] = size(peak1);

m=2;

for i = 2:B

ifpeak1(i)~=0 && peak1(i-1)==0peak2(m) = i;

m = m+1;

end

endpeak2(m) = breadth;

end

%it scans the different identified peaks of the given array and identifies

%the highest peak

function [ max_val ] = mpeak( input_var, peak_array )

max_temp = -999999;

for i=1:(length(peak_array)-1)

for j=peak_array(i):peak_array(i+1)

ifmax(input_var(peak_array(i):peak_array(i+1)))>max_temp

max_temp = max(input_var(peak_array(i):peak_array(i+1)));

max_peak = [peak_array(i) peak_array(i+1)];end

end

end

max_temp2 = -999999;

for i = max_peak(1):max_peak(2)

ifinput_var(i) > max_temp2

max_temp2 = input_var(i);

max_val = i;

end

end


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