FRACTIONAL FOURIER TRANSFORM AND ITS APPLICATIONS Major-Project Report by Alex John Koshy B050326EC Nidhin Chandran A K B050160EC Subin B B050173EC Vinay N K B050032EC Under the guidance of Dr. G. Abhilash In Partial Fulfillment of the Requirements for the Degree of Bachelor of Technology DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY, CALICUT Kerala, India April 2009
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FRACTIONAL FOURIER TRANSFORM AND
ITS APPLICATIONS
Major-Project Report
by
Alex John Koshy B050326EC
Nidhin Chandran A K B050160EC
Subin B B050173EC
Vinay N K B050032EC
Under the guidance of
Dr. G. Abhilash
In Partial Fulfillment of the Requirements
for the Degree of
Bachelor of Technology
DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY, CALICUT
Kerala, India
April 2009
i
NATIONAL INSTITUTE OF TECHNOLOGY CALICUT
DEPARTMENT OF ELECTRONICS AND COMMUNICATION
ENGINEERING
CERTIFICATE
This is to certify that this report titled FRACTIONAL FOURIER TRANS-
FORM AND ITS APPLICATIONS is a bona fide record of the major-project
done by Alex John Koshy (Roll No. B050326EC), Nidhin Chandran A K
(Roll No. B050160EC), Subin B (Roll No. B050173EC) and Vinay N K
(Roll No. B050032EC), in partial fulfillment of the requirements for the award
of Degree of Bachelor of Technology in Electronics and Communication Engineering
from National Institute of Technology, Calicut.
Dr. G. Abhilash Dr. Lillykutty Jacob
(Project Advisor) Professor and Head
Assistant Professor
29 April 2009
NIT Calicut
ii
ACKNOWLEDGEMENT
We would like to thank Dr. G.Abhilash, Assistant Professor, Department of Elec-
tronics and Communication Engineering for his guidance and inspiration in helping
us complete this project. We are also grateful to Dr. Lillykutty Jacob, Professor
and Head, Department of Electronics and Communication Engineering for providing
us with this opportunity to work on our project and also for permitting access to the
required facilities. We would also like to thank the lab staff for their technical support
and providing us assistance. We also thank our batch mates who had supported us
and provided us with greatly appreciated technical and non-technical aid throughout
our project.
Alex John Koshy
Nidhin Chandran A K
Subin B
Vinay N K
iii
Abstract
Signals can be viewed from different perspectives using different transforms. The
Fourier transform which allows us to observe the signal in terms of different fre-
quency components is widely used in signal processing and communication. Frac-
tional Fourier Transform (FrFT) is a generalized Fourier transform which allows us
to take transforms of fractional order also.
Theory of FrFT is developed from Fourier transform. For computation we move from
continuous to discrete domain and theory of Discrete Fractional Fourier Transform
(DFrFT) is discussed. A time efficient algorithm for DFrFT is also discussed. Theory
of designing optimal filter using FrFT is developed from first principles. Optimal
filters are designed for some special cases and their performance is analyzed. Some
applications other than filtering is also discussed.
%...............multiplying with chirp..................
for p=1:2*N-1
f(p)= f(p)*chirp(p);
end
%................convoluting with chirp.................
c = pi/N/sinalpha/4;
chirp2=exp(i*c*(-(2*N):2*N-1)’.^2);
Frft = conv(chirp2,f);
%............post multiplying with chirp................
Frft = chirp.*Frft(2*N:4*N-1);
%............multipling by the gain term...............
Frft = Frft(1:2*N)*exp(-i*pi*sign(sinalpha)/4
+i*alpha/2)/sqrt(abs(sinalpha))/2/sqrt(N);
%..........down sampling Frft to N terms..............
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% length(frft)
frft2=Frft(1:2:2*N);
3.2.2 Fractional Fourier Transform of Delta function
Fractional Fourier Transform of Delta function for angles 22.50, 450, 67.50, 900 are as
shown in figure 3.1 and 3.2
.
Figure 3.1: (a)Delta Function Input (b)FrFT at 22.50 (c)FrFT at 450
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Figure 3.2: (a) FrFT of Delta Function at 67.50 (b)FrFT of Delta Function at 900
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3.2.3 Fractional Fourier Transform of Sine function
Fractional Fourier Transform of Sinusoidal Signal for angles 22.50, 450, 67.50, 900 are
as shown in figure 3.3 and 3.4.
Figure 3.3: (a) Sinusoidal Input, (b) FrFT at 22.50 for Sinusoidal Input (c)FrFT at450 for Sinusoidal Input
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Figure 3.4: (a) FrFT at 67.50 for Sinusoidal Input (b) FrFT at 900 for SinusoidalInput (c) Inverse FrFT
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3.2.4 Fractional Fourier Transform of rectangular function
Fractional Fourier Transform of rectangular pulse for angles 22.50, 450, 67.50, 900 are
as shown in figure 3.5 and 3.6.
Figure 3.5: (a) Rectangular Function Input (b) FrFT at 22.50 for Rectangular Func-tion Input (c) FrFT at 450 for Rectangular Function Input (d) FrFT at 67.50 forRectangular Function Input
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Figure 3.6: (a) FrFT of Rect at 900, (b) FrFT - Magnified view of (a), (c) InverseFrFT
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Chapter 4
Filtering in Fractional FourierDomain
Consider the case of two sine waves of different frequencies. They cannot be separated
in time domain using a multiplicative filter as they are completely overlapped in time
domain. But they are easily separable in frequency domain. Similarly, consider two
impulses that are shifted in time. They are inseparable using a multiplicative filter
in frequency domain. But can be separated in time domain. It gives us an intuition
that there should be signals which are separable at some particular fractional domain
but are overlapped in time and frequency domains. Such signals can be filtered using
FrFT.
4.1 Filtering in Fractional Fourier Domain
Our scheme of FrFT filtering is as follows.
1. Analyze the signal and obtain the fractional domain at which signals get sepa-
rated.
2. Take the FrFT of the signal.
3. Remove the interfering noise part using a multiplicative window.
4. Take inverse FrFT
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The above scheme is applied in the case of Gaussian window contaminated with chirp
noise. Signals after each processing are shown. Notice that there are disturbances at
either side of the processing window because of the finite time duration of the window
we considered. See Fig. 4.1, Fig. 4.2 and Fig. 4.3
Figure 4.1: (a) Gaussian Signal (b) Chirp noise
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Figure 4.2: (a) Gaussian Signal with chirp noise (b) FrFT of a at 720 (c) Part of FrFTcorresponding to chirp
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Figure 4.3: (a) After windowing out FrFT corresponding to chirp (b) Chirp extractedby inverse FrFT (c) Gaussian extracted by inverse FrFT of (a)
27
4.2 Optimal Filter Design
Consider a case where signals that we consider are overlapped in all the fractional
domains. In this case we have to choose a window which gives maximum separation
between the signals we consider [9]. Our modified scheme of FrFT filtering will be as
follows.
1. Take the FrFT of the filter in all Fractional domains.
2. Obtain the optimum multiplicative filter in all the fractional domains.
3. Effect filtering in all the fractional domains with corresponding window.
4. Calculate SNR obtained in each order.
5. Select the fractional order that gives highest SNR.
6. Take the inverse FrFT corresponding to that order which maximizes SNR.
It is impractical to consider all the infinitesimally separated fractional orders. So we
discretize the fractional orders and searching is carried out over that set only. A trade
off is possible between processing time and optimality of fractional order. Another
method is to go for a coarse and fine searching.
Let x(t) be the transmitted signal and y(t) be the received signal after the effect of
noise n(t), such that,
y(t) = x(t) + n(t) (4.1)
We use a multiplicative filter g(t), such that we get an estimate of x(t), from y(t).
x̂(t) = g(t).y(t) (4.2)
Our aim is to find out an optimal filter g(t) which minimizes J , where J is,
J = E
[∫ ∞−∞
∣∣(x(t)− g(t).y(t))2∣∣ dt] (4.3)
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At any instant, Let x be the transmitted value and y be the received value, assuming
channel delay to be zero.
y = x+ n (4.4)
Even if x is a constant value, y will be a set of values because of n. Let x be the
ensemble of x.
i.e., x = [x1, x2, ..., xn]T such that, x1 = x2 = ..... = xn = x = x(t)t=t0
The ensemble of received values,
y = [y1, y2, ..., yn]T (4.5)
Now, x̂, the estimate of x will be,
x̂ = g.y, such that g = g(t)t=t0 .
Our aim is to minimize the norm of j = x-g.y, which happens when j ⊥ y. i.e.,
(x− g.y)T.y = 0
xT.y − g.yT.y = 0
g = (xT.y)/(yT.y) (4.6)
g(t)t=t0 =Rxy
Ryy t=t0
(4.7)
We can generalize the result to fractional domains also as,
ga(ta)ta=ta0=Rxaya
Ryaya ta=ta0
(4.8)
For each a value we find out a J value and we select the value of a which minimizes J .
29
4.2.1 Example 1 : Chirp signal contaminated with White
Gaussian Noise.
Consider the case of a Gaussian pulse contaminated with chirp noise. Gaussian will
remain Gaussian in all the fractional domains, while chirp forms an impulse at a
particular fractional order. Now we can separate the impulse corresponding to the
chirp with a rectangular window. Taking inverse FrFT, we get both signals separated.
See Fig.4.4 and Fig.4.5.
Here input is chirp x(t) and output is y(t).
y(t) = x(t) + n(t) (4.9)
In general,
ya(t) = xa(t) + na(t) (4.10)
We have seen that the optimum filter in fractional domain is of the form
gopt(ta) =Rxy(t1, t2)
Ryy(t1, t2)|t1=t2=ta (4.11)
gopt(ta) =E[xa(ta)y
∗a(ta)]
E[ya(ta)y∗a(ta)](4.12)
White noise will remain white in all the fractional domains.
E[na(ta)] = 0 (4.13)
Expanding ya(ta) , we get
gopt(ta) =[xa(ta)x
∗a(ta)]
[xa(ta)x∗a(ta)] + E[na(ta)n∗a(ta)](4.14)
Where E[na(ta)n∗a(ta)] is the average noise power.
Multiplicative filtering is carried out for different values of a and the one gives maxi-
mum SNR is selected for filtering.
30
Figure 4.4: (a) Chirp Signal (b)WGN added to Chirp Signal SNR = -6dB (c) Frac-tional Fourier Transform of the signal at −82.50s
31
Figure 4.5: (a)Optimum Multiplicative Filter (b)Extracted Chirp Signal SNR = 13dB(c) Variation of output noise power with fractional order
32
4.2.1.1 Observations
• Rectangular window is taken for simplicity. It need not be the window that
maximizes the separation.
• Error is mainly concentrated at the ends of the frame taken. This is because
of finite time width of the DFrFT window and also because of the rectangular
window filtering.
4.2.1.2 Matlab Code
%chirp in wgn.
%white noise remain white in all fractional domains.
Figure 4.8: (a) Transmitted chirp signal (b) Received signal-Multi Path, WGN SNR= -6 dB (c) Signal after FrFT for coefficients 1, 0.8, 0.6 Angle = −800
43
for k = 1:chirp_len
rcv(k) = rcv(k) + 1 * chirp(k);
rcv(200+k) = rcv(200+k) + .8 * chirp(k);
rcv(450+k) = rcv(450+k) + .6 * chirp(k);
end
plot(real(rcv));
plt = 1;
rcv_pow = sum(abs(rcv) .* abs(rcv));
%------------generating wgn----------------------
n = wgn(1,tot_len,2.6);
plot(n);
plt =1;
nos_pow = sum(abs(n) .* abs(n));
SNR = 10 * log(rcv_pow / nos_pow)
rcv = rcv + n;
plot(real(rcv));
plt = 1;
%----------search for optimum angle--------------
for ord = -90:1:90
rcv_fr = frft(rcv,ord/90);
plot(abs(rcv_fr));
plt =1;
ord
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end
rcv_fr = frft(rcv, -80/90);
plot(abs(rcv_fr));
plt=0;
peak = 0;
magn_rcv_fr = abs(rcv_fr);
peak = max(magn_rcv_fr);
threshold = .4* peak;
for k =1:tot_len
if magn_rcv_fr(k) < threshold
magn_rcv_fr(k) = 0;
rcv_fr(k) = 0;
end
end
plot(abs(rcv_fr));
plt=0;
val = 0;
rcv_fr_est = zeros(1,tot_len);
rcv_fr_est(485:495) = rcv_fr(485:495);
plot(abs(rcv_fr_est));
plt = 0;
45
rcv_est = frft(rcv_fr_est, 80/90)’;
plot (real(rcv_est));
plt = 0;
4.3.3 FrFT for measuring the acceleration of a moving object
in radial direction
When illuminated by a constant frequency sinusoid, the reflection from a radially
accelerating object will be a chirp. A search in different fractional domains is carried
out for obtaining the peak. Let αmax be the angle corresponding to maximum peaking
[5].
Transmitted signal
St(t) = ej2πf0t (4.19)
Received signal
Sr(t) = ej2πf0t+j2π
2vλ0t+π 2a
λ0t2+Φ0 (4.20)
The estimation of radial acceleration is
aest = −λ0fs2T
cotαmax (4.21)
46
Chapter 5
Critical Evaluation and Conclusion
5.1 Critical Evaluation
Based on the study and analysis of the simulation results, we conclude that:
1. FrFT gives one more degree of freedom while designing signal processing tools
compared to Fourier transform.
2. Signal analysis in time-frequency plane is easy with the help of FrFT.
3. FrFT is computationally efficient and has the same order of complexity as that
of Fourier transform.
4. FrFT based computations are to be done frame-wise, which will result in errors
due to windowing and finite time duration. It can be reduced to a great extent
by taking frames with 50 percent overlap.
5. Most of the time, optimum order for computation will be unknown, as a result,
a search over a range of fractional orders should be carried out, which is a time
consuming task.
6. FrFT performs excellently if any of the signals that we consider is chirp-like.
7. Signals of non-stationary spectral characteristics can be analyzed using FrFT
with superior performance compared to Fourier transform.
47
8. For speech and music processing, advantage that we get by using FrFT is min-
imal.
9. Signals after FrFT processing will contain complex part, which should be con-
sidered for any further processing.
5.2 Conclusion
We can conclude that FrFT is a more general method for signal processing and sys-
tem design. FrFT based systems can replace the current frequency domain systems.
FrFT can have bigger roles in fields like radar and sonar where chirp signals are very
common. There is large scope of research in FrFT in the fields of spread spectrum
communication, signal watermarking and encryption, cognitive radio and so on. FrFT
can also be used for the design of faster optical signal processing systems.
48
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