How to solve intergal for analytical form of ambiguity function

Hi,

I am new to this subject and I need some advice or hint to solve the following problem. I have to develop a matlab program were I determine the analytical form of the ambiguity function with an rectangualr pulse, like in the picture.

My little program looks like this

syms t T v tau
assume(abs(tau) <= T)
func = rectangularPulse(t)*rectangularPulse(t-tau)*exp(2i*pi*v*t);

F = int(func,t,-inf,inf)

where I put all together, but the solution is

F =
 
piecewise((imag(v) < 0 & ~0 < v*1i | 0 < imag(v)) & tau == 0, (exp(-pi*v*1i)*1i)/(2*v*pi) - (exp(pi*v*1i)*1i)/(2*v*pi), (1 <= tau | tau <= -1) & (imag(v) < 0 & ~0 < v*1i | 0 < imag(v)), 0, (imag(v) < 0 & ~0 < v*1i | 0 < imag(v)) & tau in Dom::Interval([0], [1]), (exp(pi*v*(tau - 1/2)*2i)*1i)/(2*v*pi) - (exp(pi*v*1i)*1i)/(2*v*pi), (imag(v) < 0 & ~0 < v*1i | 0 < imag(v)) & tau in Dom::Interval([-1], [0]), - (exp(pi*v*(tau + 1/2)*2i)*1i)/(2*v*pi) + (exp(-pi*v*1i)*1i)/(2*v*pi), (0 <= imag(v) | ~0 < v*1i) & imag(v) <= 0 & (imag(v) < 0 & ~0 < v*1i | 0 < imag(v)) & (0 <= imag(v) | 0 < v*1i) & 1 <= tau, int(exp(pi*t*v*2i), t, piecewise(0 <= tau, tau + 1/2, tau <= 0, 1/2), Inf) + int(-exp(pi*t*v*2i), t, piecewise(-1 <= tau, tau + 1/2, tau <= -1, -1/2), Inf), (0 <= imag(v) | ~0 < v*1i) & imag(v) <= 0 & (imag(v) < 0 & ~0 < v*1i | 0 < imag(v)) & (0 <= imag(v) | 0 < v*1i) & tau <= -1, int(exp(pi*t*v*2i), t, piecewise(0 <= tau, tau - 1/2, tau <= 0, -1/2), Inf) + int(-exp(pi*t*v*2i), t, piecewise(1 <= tau, tau - 1/2, tau <= 1, 1/2), Inf) - (exp(-pi*v*1i)*1i)/(2*v*pi) + (exp(pi*v*1i)*1i)/(2*v*pi), (0 <= imag(v) | ~0 < v*1i) & imag(v) <= 0 & (imag(v) <= 0 | 0 < tau) & (imag(v) < 0 & ~0 < v*1i | 0 < imag(v)) & (0 <= imag(v) | 0 < v*1i) & tau in Dom::Interval([0], [1]) & (0 <= imag(v) | 0 < v*1i | 0 < tau) & (0 <= imag(v) | ~0 < v*1i | 0 < tau), int(exp(pi*t*v*2i), t, piecewise(0 <= tau, tau + 1/2, tau <= 0, 1/2), Inf) + int(-exp(pi*t*v*2i), t, piecewise(-1 <= tau, tau + 1/2, tau <= -1, -1/2), Inf) + (exp(pi*v*(tau - 1/2)*2i)*1i)/(2*v*pi) - (exp(pi*v*1i)*1i)/(2*v*pi), (0 <= imag(v) | ~0 < v*1i) & imag(v) <= 0 & (imag(v) <= 0 | tau < 0) & (imag(v) < 0 & ~0 < v*1i | 0 < imag(v)) & (0 <= imag(v) | 0 < v*1i) & tau in Dom::Interval([-1], [0]) & (0 <= imag(v) | 0 < v*1i | tau < 0) & (0 <= imag(v) | ~0 < v*1i | tau < 0), int(exp(pi*t*v*2i), t, piecewise(0 <= tau, tau - 1/2, tau <= 0, -1/2), Inf) + int(-exp(pi*t*v*2i), t, piecewise(1 <= tau, tau - 1/2, tau <= 1, 1/2), Inf) - (exp(pi*v*(tau + 1/2)*2i)*1i)/(2*v*pi) + (exp(pi*v*1i)*1i)/(2*v*pi), (0 <= imag(v) | ~0 < v*1i) & imag(v) <= 0 & (imag(v) <= 0 | 0 < tau) & (imag(v) <= 0 | 1 < tau) & (imag(v) <= 0 | tau < 1) & (imag(v) < 0 & ~0 < v*1i | 0 < imag(v)) & (0 <= imag(v) | 0 < v*1i) & 0 <= tau & (0 <= imag(v) | 0 < v*1i | 0 < tau) & (0 <= imag(v) | 0 < v*1i | 1 < tau) & (0 <= imag(v) | 0 < v*1i | tau < 1) & (0 <= imag(v) | ~0 < v*1i | 0 < tau) & (0 <= imag(v) | ~0 < v*1i | 1 < tau) & (0 <= imag(v) | ~0 < v*1i | tau < 1), int(exp(pi*t*v*2i), t, piecewise(0 <= tau, tau - 1/2, tau <= 0, -1/2), Inf) + int(-exp(pi*t*v*2i), t, piecewise(1 <= tau, tau - 1/2, tau <= 1, 1/2), Inf), (0 <= imag(v) | ~0 < v*1i) & imag(v) <= 0 & (imag(v) <= 0 | tau < 0) & (imag(v) <= 0 | -1 < tau) & (imag(v) <= 0 | tau < -1) & (imag(v) < 0 & ~0 < v*1i | 0 < imag(v)) & (0 <= imag(v) | 0 < v*1i) & tau <= 0 & (0 <= imag(v) | 0 < v*1i | tau < 0) & (0 <= imag(v) | 0 < v*1i | -1 < tau) & (0 <= imag(v) | 0 < v*1i | tau < -1) & (0 <= imag(v) | ~0 < v*1i | tau < 0) & (0 <= imag(v) | ~0 < v*1i | -1 < tau) & (0 <= imag(v) | ~0 < v*1i | tau < -1), int(exp(pi*t*v*2i), t, piecewise(0 <= tau, tau + 1/2, tau <= 0, 1/2), Inf) + int(-exp(pi*t*v*2i), t, piecewise(-1 <= tau, tau + 1/2, tau <= -1, -1/2), Inf) + (exp(-pi*v*1i)*1i)/(2*v*pi) - (exp(pi*v*1i)*1i)/(2*v*pi), (imag(v) <= 0 | 0 < tau) & (imag(v) <= 0 | tau < 0) & (imag(v) <= 0 | -1 < tau) & (imag(v) <= 0 | 1 < tau) & (imag(v) <= 0 | tau < -1) & (imag(v) <= 0 | tau < 1) & in(tau, 'real') & (0 <= imag(v) | 0 < v*1i | 0 < tau) & (0 <= imag(v) | 0 < v*1i | tau < 0) & (0 <= imag(v) | 0 < v*1i | -1 < tau) & (0 <= imag(v) | 0 < v*1i | 1 < tau) & (0 <= imag(v) | 0 < v*1i | tau < -1) & (0 <= imag(v) | 0 < v*1i | tau < 1) & (0 <= imag(v) | ~0 < v*1i | 0 < tau) & (0 <= imag(v) | ~0 < v*1i | tau < 0) & (0 <= imag(v) | ~0 < v*1i | -1 < tau) & (0 <= imag(v) | ~0 < v*1i | 1 < tau) & (0 <= imag(v) | ~0 < v*1i | tau < -1) & (0 <= imag(v) | ~0 < v*1i | tau < 1), int(exp(pi*t*v*2i), t, piecewise(0 <= tau, tau - 1/2, tau <= 0, -1/2), Inf) + int(exp(pi*t*v*2i), t, piecewise(0 <= tau, tau + 1/2, tau <= 0, 1/2), Inf) + int(-exp(pi*t*v*2i), t, piecewise(-1 <= tau, tau + 1/2, tau <= -1, -1/2), Inf) + int(-exp(pi*t*v*2i), t, piecewise(1 <= tau, tau - 1/2, tau <= 1, 1/2), Inf))

and I don't understand what's exactly going on here. To me this seems that there are different solutions for a different constraints of tau, right? But in F I've not found the solution given in the picture above.

Can someone please help with this?

regards

[Walter Roberson]

Are you able to restrict tau or v to be real-valued? That would simplify the outcome.

[August Lamm]

Thank you for your fast reply. Now you ask, I think 3 variables must be real except of v, because it is in the exponent of the e-function. And if I do the calculations with v as real I get NaN as solution. So, I modded the rect function a litle bit and now the code looks like this

syms t tau T real
syms v 
assume(abs(tau) <= T)

func = 1/sqrt(T)*rectangularPulse(-T/2,T/2,t)*1/sqrt(T)*rectangularPulse(-T/2,T/2,t-tau)*exp(2i*pi*v*t);

F = int(func,t,-inf,inf)

but the solution now is also a long term

F =
 
piecewise(0 < imag(v) & tau == 0, (exp(-pi*T*v*1i)*1i)/(2*T*v*pi) - (exp(pi*T*v*1i)*1i)/(2*T*v*pi), 0 < imag(v) & 0 <= T + tau & tau <= 0, (exp(-pi*T*v*1i)*1i)/(2*T*v*pi) - (exp(v*pi*(T/2 + tau)*2i)*1i)/(2*T*v*pi), 0 < imag(v) & tau <= T & ~in(v, 'real') & 0 <= tau, (exp(-v*pi*(T/2 - tau)*2i)*1i)/(2*T*v*pi) - (exp(pi*T*v*1i)*1i)/(2*T*v*pi), imag(v) < 0 & 0 < v*1i & tau == 0, (- (Inf*1i)/v + (exp(-pi*T*v*1i)*1i)/(2*v*pi))/T - (- (Inf*1i)/v + (exp(pi*T*v*1i)*1i)/(2*v*pi))/T, imag(v) < 0 & ~0 < v*1i & tau == 0, - ((limit(exp(pi*t*v*2i), t, Inf)*1i)/(2*v*pi) - (exp(-pi*T*v*1i)*1i)/(2*v*pi))/T + ((limit(exp(pi*t*v*2i), t, Inf)*1i)/(2*v*pi) - (exp(pi*T*v*1i)*1i)/(2*v*pi))/T, imag(v) < 0 & 0 < v*1i & 0 <= T + tau & tau <= 0, - (- (Inf*1i)/v + (exp(v*pi*(T/2 + tau)*2i)*1i)/(2*v*pi))/T + (- (Inf*1i)/v + (exp(-pi*T*v*1i)*1i)/(2*v*pi))/T, imag(v) < 0 & ~0 < v*1i & 0 <= T + tau & tau <= 0, ((limit(exp(pi*t*v*2i), t, Inf)*1i)/(2*v*pi) - (exp(v*pi*(T/2 + tau)*2i)*1i)/(2*v*pi))/T - ((limit(exp(pi*t*v*2i), t, Inf)*1i)/(2*v*pi) - (exp(-pi*T*v*1i)*1i)/(2*v*pi))/T, imag(v) < 0 & tau <= T & 0 < v*1i & ~in(v, 'real') & 0 <= tau, (- (Inf*1i)/v + (exp(-v*pi*(T/2 - tau)*2i)*1i)/(2*v*pi))/T - (- (Inf*1i)/v + (exp(pi*T*v*1i)*1i)/(2*v*pi))/T, imag(v) < 0 & tau <= T & ~0 < v*1i & ~in(v, 'real') & 0 <= tau, ((exp(-v*pi*(T/2 - tau)*2i)*1i)/(2*v*pi) - (limit(exp(pi*t*v*2i), t, Inf)*1i)/(2*v*pi))/T + ((limit(exp(pi*t*v*2i), t, Inf)*1i)/(2*v*pi) - (exp(pi*T*v*1i)*1i)/(2*v*pi))/T, T <= tau & ~in(v, 'real') | 0 < imag(v) & in(tau, 'real') & T + tau <= 0 | imag(v) < 0 & in(tau, 'real') & T + tau <= 0, 0, (0 <= imag(v) | ~0 < v*1i) & imag(v) <= 0 & (imag(v) <= 0 | 0 < tau) & ~in(v, 'real') & (imag(v) <= 0 | T < tau) & (imag(v) <= 0 | tau < T) & (0 <= imag(v) | 0 < v*1i) & 0 <= tau & (0 <= imag(v) | T < tau | ~0 < v*1i) & (0 <= imag(v) | tau < T | ~0 < v*1i) & (0 <= imag(v) | 0 < v*1i | 0 < tau) & (0 <= imag(v) | T < tau | 0 < v*1i) & (0 <= imag(v) | tau < T | 0 < v*1i) & (0 <= imag(v) | ~0 < v*1i | 0 < tau), int(-exp(pi*t*v*2i)/T, t, piecewise(T <= tau, tau - T/2, tau <= T, T/2), Inf) + int(exp(pi*t*v*2i)/T, t, piecewise(0 <= tau, tau - T/2, tau <= 0, -T/2), Inf), (imag(v) <= 0 | 0 < T + tau) & (imag(v) <= 0 | T + tau < 0) & T <= tau & (imag(v) <= 0 | 0 < tau) & (imag(v) <= 0 | tau < 0) & ~in(v, 'real') & (0 <= imag(v) | 0 < v*1i | 0 < T + tau) & (0 <= imag(v) | 0 < v*1i | T + tau < 0) & (0 <= imag(v) | 0 < v*1i | 0 < tau) & (0 <= imag(v) | 0 < v*1i | tau < 0) & (0 <= imag(v) | ~0 < v*1i | 0 < T + tau) & (0 <= imag(v) | ~0 < v*1i | T + tau < 0) & (0 <= imag(v) | ~0 < v*1i | 0 < tau) & (0 <= imag(v) | ~0 < v*1i | tau < 0), int(-exp(pi*t*v*2i)/T, t, piecewise(0 <= T + tau, T/2 + tau, T + tau <= 0, -T/2), Inf) + int(exp(pi*t*v*2i)/T, t, piecewise(0 <= tau, T/2 + tau, tau <= 0, T/2), Inf), (imag(v) <= 0 | 0 < T + tau) & (imag(v) <= 0 | T + tau < 0) & (imag(v) <= 0 | 0 < tau) & (imag(v) <= 0 | tau < 0) & in(tau, 'real') & (imag(v) <= 0 | T < tau) & (imag(v) <= 0 | tau < T) & (0 <= imag(v) | T < tau | ~0 < v*1i) & (0 <= imag(v) | tau < T | ~0 < v*1i) & (0 <= imag(v) | 0 < v*1i | 0 < T + tau) & (0 <= imag(v) | 0 < v*1i | T + tau < 0) & (0 <= imag(v) | 0 < v*1i | 0 < tau) & (0 <= imag(v) | 0 < v*1i | tau < 0) & (0 <= imag(v) | ~0 < v*1i | 0 < T + tau) & (0 <= imag(v) | ~0 < v*1i | T + tau < 0) & (0 <= imag(v) | T < tau | 0 < v*1i) & (0 <= imag(v) | tau < T | 0 < v*1i) & (0 <= imag(v) | ~0 < v*1i | 0 < tau) & (0 <= imag(v) | ~0 < v*1i | tau < 0), int(-exp(pi*t*v*2i)/T, t, piecewise(0 <= T + tau, T/2 + tau, T + tau <= 0, -T/2), Inf) + int(-exp(pi*t*v*2i)/T, t, piecewise(T <= tau, tau - T/2, tau <= T, T/2), Inf) + int(exp(pi*t*v*2i)/T, t, piecewise(0 <= tau, tau - T/2, tau <= 0, -T/2), Inf) + int(exp(pi*t*v*2i)/T, t, piecewise(0 <= tau, T/2 + tau, tau <= 0, T/2), Inf))

[Walter Roberson]

Run it under multiple conditions,

assume(imag(v) < 0)
Fim_neg = simplify(F)
assume(v, 'real')
Fim_zero = simplify(F)
assume(imag(v) > 0)
Fim_pos = simplify(F)
syms z    %resets assumptions

Also, you might want to break it up into tau negative or tau positive

[August Lamm]

I've tried this way, but could not get the wanted result out of it.

cheers

Answer 1

[Paul]

I was not able to get Matlab to return a solution in the expected form of the answer. But we can show that Matlab's solution is the same as the expected result.

syms t tau v w T real
s(t) = rectangularPulse(-T/2,T/2,t)/sqrt(T);

Note that the ambiguity function is essentially the inverse Fourier transform of s(t)*s(t-tau) with t and w reversed.

F1(tau,w,T) = ifourier(s(t)*s(t-tau),t,w)*2*sym(pi);  % simpler form?
F1(tau,v,T) = F1(tau,2*sym(pi)*v,T)

F1(tau, v, T) = 

Compute the ambiguity function with the defining integral.

F2(tau,v,T) = int(s(t)*s(t-tau)*exp(1j*2*sym(pi)*v*t),t,-T,T) % only need to integrate from outside the limits of s(t)

F2(tau, v, T) = 

Ambiguity function as defined in the question.

F3 = @(tau,v,T) (exp(1j*pi.*v.*tau).*(1-abs(tau)./T).*sinc(v.*(T-abs(tau))).*(abs(tau)<=T)); % expected answer

Evaluate and compare all three for T = 1.

Tval = 1;
tauvec = -1.5:.1:1.5;
vvec   = -5:.1:5;
vvec(vvec==0) = NaN;  % avoid divide by 0 in F1 and F2
[Tau,V] = meshgrid(tauvec,vvec);

F1val = double(vpa(F1(Tau,V,Tval)));
F2val = double(vpa(F2(Tau,V,Tval)));
F3val = double((F3(Tau,V,Tval)));

figure
surf(Tau,V,abs(F1val));xlabel('tau');ylabel('v');

figure
surf(Tau,V,abs(F2val));xlabel('tau');ylabel('v');

figure
surf(Tau,V,abs(F3val));xlabel('tau');ylabel('v');

figure
surf(Tau,V,angle(F1val));xlabel('tau');ylabel('v');

figure
surf(Tau,V,angle(F2val));xlabel('tau');ylabel('v');

figure
surf(Tau,V,angle(F3val));xlabel('tau');ylabel('v');

figure
surf(Tau,V,abs(F1val - F2val));xlabel('tau');ylabel('v');

figure
surf(Tau,V,abs(F1val - F3val));xlabel('tau');ylabel('v');

figure
surf(Tau,V,abs(F2val - F3val));xlabel('tau');ylabel('v');

Code took too long to run; the figures aren't shown.

[August Lamm]

Thank you for your loooong, expansive answer and work. :) For me the code was running fine and give me 9 nice plots and I have now some questions about it. Please don't get me wrong, I don't want to bash your results or something. I am only courious about it. And sorry If I am asking silly questions, because I am not one hell of a math guy.

First, in this line you typed the wanted outcome

F3 = @(tau,v,T) (exp(1j*pi.*v.*tau).*(1-abs(tau)./T).*sinc(v.*(T-abs(tau))).*(abs(tau)<=T)); % expected answer

, but I think in the sinc function is a "pi" missing or is there a reason why it is not there? Also I don't understand the last multiplication

.*(abs(tau)<=T)

, because why is it a multiplication in the first place? I thought of "abs(tau)<=T" it as a resctriction for the whole ambiguity function or am I completely wrong?

How can the plots prove that the found results are equal to the wanted answer when tne subtractions are not zero? For my understanding if they are not completely the same the plots are unequal to 0 like in the plots 8

figure
surf(Tau,V,abs(F1val - F3val));xlabel('tau');ylabel('v');

and 9

figure
surf(Tau,V,abs(F2val - F3val));xlabel('tau');ylabel('v');

But they look the same.

How do you get these nice latex like equation outputs?

cheers

[Paul]

I'm using the sinc() function in the Signal Processing Toolbox. It multiplies the argument by pi internally. Maybe you're using a different sinc() function, which could explain why you don't get the expected results in the plots? Check with which()

>> which sinc -all
C:\Program Files\MATLAB\R2019a\toolbox\signal\signal\sinc.m
C:\Program Files\MATLAB\R2019a\toolbox\symbolic\symbolic\@sym\sinc.m  % sym method
C:\Program Files\MATLAB\R2019a\toolbox\signal\signal\@tall\sinc.m     % tall method

The multiplication by (abs(tau) <= T) was to match the definition of X(tauv,v) in the Question that showed " = 0 elsewhere." I don't think there is a restriction on the ambiguity function, i.e, I think its domain covers -inf < tau < inf, but I could be wrong.

I'm not sure what you're seeing in the plots. When I run the code (in 2019a), here is what I get for Figure 7:

And Figure 8:

The Symbolic Math Toolbox can generate Latex expressions

doc latex

though in this forum I just run the code and the website does it automatically.

[August Lamm]

Your guess was right. I added a pi to the argument of the sinc function udn that gave me the wrong plot. Now, after removing the pi I also have the same plot as you.