# Equation visualization of complex sinusoid function

I'm trying to learn Fourier Transform & Signal

I can visualize how this expression $A \sin(2 \pi f t)$ could turn into that curvy sinusoidal signal. But i couldn't visualize how $F(\nu)e^{2 \pi i f t}$ would look like as a signal....

Why the natural exponent suddenly popped up? what does it do? i understand that $F(\nu)$ gives information of the amplitude and the phase shift, but what sort of information i could get from $e$ ? Does all signals always decaying/growing by time?

I saw this a lot in Laplace Transform: $e^{-st}$ but don't understand how it would look like as a signal.

The exponent function is directly related to the sine and cosine via Euler equation:

$$e^{i \theta} = \cos(\theta) + i \sin(\theta)$$

For example for function $z=e^{i \pi t}$:

$$z=\cos(\pi t) + i \sin(\pi t)$$

You could plot it in 3D, where x axis is the time, and y & z axes are real and imaginary part of your complex signal, respectively. In 3D that is going to be a helix: But if you look at projections onto the y=0 and z=0 planes, then you will notice sine: and cosine waves: On the other hand, if you look at the x=0 projection, then you will observe a (unit) circle: You can find the applet with this visualization in Wolfram Demonstration Project.

• A note on the projections: The sine/cosine and circle are not perfectly visible, since there is some perspective distortion in the images. However, if the software would have rendered it without perspective, the three functions would be clearly visible. – M529 Mar 21 '16 at 11:58
• Thank you for the great illustration + explanation. It helps a lot! And i'm sorry for succumbing to desperation and abandoned signal subject for a long time... – Suwandy Oct 25 '18 at 14:25

You can use fft function and plot it, I didn't know if you're already try that:

NFFT = 2^nextpow2(L); % Next power of 2 from length of y
Y = fft(y,NFFT)/L;
f = Fs/2*linspace(0,1,NFFT/2+1);

% Plot single-sided amplitude spectrum.
plot(f,2*abs(Y(1:NFFT/2+1)))
title('Single-Sided Amplitude Spectrum of y(t)')
xlabel('Frequency (Hz)')
ylabel('|Y(f)|')