Frequency Response of 3-Point Averaging System

[tenacity2986]

Homework Statement

y[n]= (x[n]+x[n-1]+x[n-2]) / 3 is the input output relationship

Homework Equations

Find the Frequency response.

The Attempt at a Solution

Ok I am very aware that I can easily find the impulse response and graph this and I can even get the general format for the frequency response, however how doyou figure out the fundamental period? I know its 2pi/T, where T is the fund. period but I don't see how I can graph x[n], y[n], or find the period

 

[Redbelly98]

I don't see why you need the fundamental period (whatever that means in this situation). Sounds like they just want the amplitude, and possibly the phase, of the output given a unit-amplitude, sinusoidal input.

 

[piercas]

from the given input output relationship.

In order to find the frequency response of this 3-point averaging system, we can use the Z-transform. The Z-transform is a mathematical tool that converts a discrete-time signal into a complex frequency domain representation. In this case, we can use the Z-transform to find the transfer function of the system, which will give us the frequency response.

First, we can rewrite the input-output relationship in terms of the Z-transform as:

Y(z) = (X(z) + z^-1X(z) + z^-2X(z)) / 3

Then, we can solve for the transfer function H(z) by dividing both sides by X(z):

H(z) = Y(z) / X(z) = (1 + z^-1 + z^-2) / 3

Next, we can plot the magnitude and phase of the transfer function in the frequency domain by substituting z = e^(jω), where ω is the frequency in radians per sample. This will give us the frequency response of the system.

Finally, to find the fundamental period, we can look at the frequency response graph and find the smallest value of ω where the magnitude of H(z) is 0. This corresponds to the fundamental period T = 2π/ω.