DynCont#5: Z transform, Phils Lab, DDorran

Phisl lab

1.  Z transform is better than Fourier transform because it can do 2 things:
1. Check the stability of the implemented difference equation
2. See the frequency response of the system, that the difference equation represents
2. Z transform based stability check: Simple, replace y[n-1] by z^-1 Y and form the Z transfer function and check the location of poles . Zeros can be out side the unit circle for stable system
3. Z transform based frequency response z^(-k) = e^(-j theta k), where theta  = 2*pi*f/fs. Maximum value f can take is fs/2 so maximum value theta can take is theta = pi. 
4. For DC gain, omega is zero so z = 1
5. For high frequency gain, z= -1

 DDorran

1. Z transform is used to detect the presence of exponentially increasing or decreasing components in the signal
2. By identifying the increasing or decreasing oscillations in the impulse response  of a system we can determine if that system is stable or unstable
3. x[n] is a signal with different values for different n. z^(-n) is also a signal for different values of z. Multiplying individual values of the 2 signals and adding them is basically taking correlation of 2 signals. Higher the correlation, more is one signal present in another signal
4. Correlation is the measure of presence of one signal in another signal.
   
5. When taking correlation of 2 signals, note that if second signal is constructed using z = 1, then we are basically taking the DC value of the first signal. Also see 4th point of previous stack.
6. Z is e^(-sT). Value of the z transform at z = 1 in the plot with z values as x axis is the DC value of the signal.