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

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SUBHO MITRA am 4 Jan. 2021
Kommentiert: SUBHO MITRA am 3 Nov. 2022
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
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);
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!
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### Antworten (1)

Gokul Nath S J am 17 Okt. 2022
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.
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SUBHO MITRA am 3 Nov. 2022
In such case, the amplitude should come exactly half of 1, in this case, 0.5.

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