why ifft is complex coefficient?

i have to multiply complex number to fft of signal

but the symmetric property is eliminated when complex number is multiply

So the ifft is also complex number

and i do multiply on half of fft

and make symmetric property using filplr and conj

but eventhough i match the symmetric property, the ifft of that frequency spectrum is complex number

code is below

why that happened?

help me

clear all;
close all;

fs = 1000;
ts = 1/ fs;
f = 200;
gain = exp(i*10);

t = 0 : ts : 0.1;
x = sin(2*pi*f*t); % original signal

X = fft(x);


X1 = X(1:51)*gain; %multiply complex number
% X1(1) = 1; %when i do this the coeffcient is real 
X2 = fliplr(X1); %make symetric

X3 = [X1 conj(X2)];
X4 = X3(1:end-1);
x4 = ifft(X4);

figure(1)
plot(x4);



X1 = X(1:51)*gain; 
X2 = fliplr(X1);

X3 = [X1 conj(X2)];
X4 = X3(1:end-1);
x4 = ifft(X4) % coefficient is complex number

figure(2)
plot(x4);

Answer 1

[Nadia Shaik]

Hi,

From my understanding you are getting the expected ifft result when the first element of the input is a real number and a different result when the first element is a complex number.

In ifft , the conjugate symmetry holds good from 2nd element to last element of the input. For a real signal, the first element of fft is always real.

So, while computing ifft, the same is expected. The required real signal can be obtained when the first element is a real value.

Hope it helps!