# Cross Correlation Between Input-Output Sine Waves

I am writing an algorithm to estimate the frequency transfer function of the system. For this, I want to use the Cross-correlation Between Input-Output Sine Waves method. There are a few things I don't understand about method: I - What does capital N mean? II - Is and Ic scalar or vector(depend on N)? These are my thoughts : I - N represents the number of outputs I collected for a specific w value with a sampling periode. II - Scalar. Could you help?

These formulas are discrete fourier transform.

It's very standard procedure when you analyze wave-alike processes. What it does it transforms the signal from the time-space to the frequencies-space. The idea is that you take sin-wave with specific frequency, multiply it by signal and if the value is big then you have something similar to the wave of this frequency.

$$\frac{1}{NT}$$ term is just overall time of the sequence (N samples with length T)

The $$I_{s}$$ and $$I_{c}$$ is amplitude(intensity) for the sin and cos, respectively. You get them for the specific frequency $$\omega$$ and it's a single number for each. From this you can get full amplitude (G) and phase $$\phi$$. Why do you need phase? Because it's important - if you have two waves with similar frequency and $$\phi = \pi$$, then they would cancel each other, and if they have same phase, they would double.

• Thank you for your helpful answer. Can you give an example for N and T for better understanding? For example, What can we say for N and T if I collect samples for y = sin (pi * t) with a sampling period of 0.1 seconds from 0 to 3 seconds? – bopele May 1 '20 at 20:36
• Than you would have N = 30 samples, T=0.1 second. – Kirill Fedyanin May 1 '20 at 20:38
• Since the upper limit of the summation must be an integer (NT = 30 * 0.1 = 3), I should select the product of N and T to be an integer right? – bopele May 1 '20 at 20:50
• Not really, what they mean by this notation I believe is that you iteratate from 0 till NT with step T. I.e. for N=30 and T = 0.1 it would be k = 0.1, 0.2, 0.3 ... 3. – Kirill Fedyanin May 1 '20 at 20:51
• But it could be from 0 to 2.5 with step 0.1 and then k = 0.1, 0.2, 0.3, ... 2.5 – Kirill Fedyanin May 1 '20 at 20:52