# Z transform of a discrete filter

• asset101
In summary, the goal of the conversation was to find the difference equation relating input u(k) and output y(k) by using the given transfer function H(z) = (1 + (1/2)z^-1)/((1 - (1/2)z^-1)(1 + (1/3)z^-1)). The solution provided was y(k) - (1/6)y(k-1) - (1/6)y(k-2) = u(k) + (1/2)y(k-1). However, the person asking for help has not been able to solve the problem using their usual method and is seeking assistance.

#### asset101

The question is to find the difference equation relating u(k) and y(k), which are input and output respectively.

H(Z) is given as

H(Z) = $$\frac{1+(1/2)z^{-1}}{(1-(1/2)z^{-1})(1+(1/3)z^{-1})}$$

Solution that is given:

y(k)-$$\frac{1}{6}$$y(k-1)-$$\frac{1}{6}$$y(k-2) = u(k)+$$\frac{1}{2}$$y(k-1)

I have had many attempts and haven't be correct. The thing is i am applying the same method as questions of a similar nature which i got correct. Any help would be appreciated.
Cheers

Ok, but what have you tried so far? All you need do is to write H(z)=Y(z)/U(z) and then express Y(z) (...) = U(z) (...) where ... refers to the appropriate expression terms of the transfer function. Then simply apply the Z-inverse transform to get the answer.

Your solution is correct. The difference equation relating u(k) and y(k) for the given Z transform is:

y(k) - (1/6)y(k-1) - (1/6)y(k-2) = u(k) + (1/2)y(k-1)

This can also be written as:

y(k) = u(k) + (1/2)y(k-1) + (1/6)y(k-2) + (1/6)y(k-3)

The reason you may have had trouble getting the correct solution is that there are multiple ways to write a difference equation for a given Z transform. It is important to carefully follow the steps and ensure that all terms are correctly accounted for. If you are still having trouble, it may be helpful to seek guidance from a colleague or a tutor.

## What is the Z transform of a discrete filter?

The Z transform of a discrete filter is a mathematical representation of the filter in the frequency domain. It is used to analyze the behavior of the filter and determine its frequency response.

## How is the Z transform related to the discrete Fourier transform (DFT)?

The Z transform is closely related to the DFT, as it can be obtained from the DFT by setting the frequency variable to the complex value z = e. This allows for a simpler representation of the filter in the frequency domain.

## What is the significance of poles and zeros in the Z transform of a discrete filter?

Poles and zeros in the Z transform represent the locations of the filter's resonances and anti-resonances, respectively. They provide insight into the filter's frequency response and can be used to design filters with specific characteristics.

## How do I interpret the magnitude and phase response of a discrete filter's Z transform?

The magnitude response of the Z transform represents the gain of the filter at different frequencies, while the phase response represents the amount of delay caused by the filter at those frequencies. Both of these responses are important in understanding the behavior of the filter.

## How is the Z transform used in digital signal processing (DSP)?

The Z transform is a crucial tool in DSP, as it allows for the analysis and design of digital filters. It is used in various applications such as audio and image processing, communications systems, and control systems.

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