Why the amplitude of the final pulses are different from the input pulse?

I have a gaussian pulse in time. I want to shift it in time. So I am putting the linear spectral phase modulation to it. But the amplitude is different from the input pulse while I take the input as Cos(wt) and Exp(iwt). please give some suggestions.

Codes are following:

t=-50:0.0001:50;

f=0.375; %centre frequency

w0=2*pi*f; %centre frequency in rad/s

t0=0;

alpha=17

T=5;

F=(-500000:500000)/((1000001)*0.0001); %frequency scale

w=2*pi*F;

m=length(w);

d=mean(diff(w));

Fs=(w(2)-w(1))*m/(2*pi);

Ts=1/Fs;

t1=Ts*(-(m)/2:(m-1)/2); %time scale after IFFT

w_ref=2*pi*0.375;

y=exp(-((t-t0)/T).^2).*cos(1*w0*(t)); %Input pulse (It is giving smaller amplitude)

% OR

%y=exp(-((t-t0)/T).^2).*exp(i*w0*(t));

y0=0.1*cos(w0*t);

Y=(fft((y)));

Y1=fftshift(Y);

M_w=exp(1i*(alpha*(w-w_ref))); %linear mask

Ew=Y.*M_w; %Modulated spectrum

Ew1=Y1.*M_w;

Et=ifft(ifftshift(Ew1)); %Shaped pulse

I=(Et).^2;

P=angle(Ew1);

P=unwrap(P); %Phase

subplot(2,2,1)

plot(t,(y))

%xlim([-0.5 4])

title('Input Pulse')

xlabel('Time')

ylabel('Intensity')

axis square

grid on

subplot(2,2,2)

yyaxis left

plot(F,abs(Ew1))

yyaxis right

plot(F,P)

xlim([0.35 0.45])

title('Modulated Spectrum')

xlabel('Frequency')

ylabel('Intensity')

axis square

grid on

subplot(2,2,3)

plot(t1,(real(Et)))

ylim([-1.1 1.1])

hold all

plot(t1,y0)

title('Temporal profile of the Shaped Pulse')

xlabel('Time')

ylabel('Intensity')

axis square

grid on

subplot(2,2,4)

plot(F,(P))

%xlim([0 50])

title('Spectral Phase of the shaped pulse')

xlabel('Freq')

ylabel('Phase')

axis square

grid on

Here you can see that the Input y as Cosine function gives smaller amplitude after appling the mask function M_w than the input y as Exp function. I want to know is it like that? pls help!

Answer 1

[Gokul Nath S J]

Dear Subho,

The amplitudes of the function won’t be same. This can be verified mathematically by expanding cosine in terms of its complex exponential. This will give you a factor of 2 in the denominator.

Moreover, as mentioned in question multiplying with a complex exponential will shift the signal. However, the amplitude scaling occurs if multiplied with a cosine or sine.

[SUBHO MITRA]

In such case, the amplitude should come exactly half of 1, in this case, 0.5.