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Having the Transfer Function of a discrete system as such:

$$H(z) = (1.011)\frac{z-0.9756}{z-0.9975}$$

How would you find the gain at ω₁=3 rad/s at a sampling time of T=0.25 seconds.

The answer to the solution is: 0.1013 dB

I'm having problems with this. I am aware that $z=e^{sT}=e^{jωT}$. I substitute the values of ω₁ and T, and it always results in a complex error, despite attempting it multiple times.

Could someone help how you would get the final solution, with steps provided?

Kind regards, Alan.

2 Answers2

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First of all, convert the Z-transform $H(z)$ into the Frequency response $H(e^{j\omega})$ by replacing $z$ with $e^{j\omega}$.

$$ \begin{align} H(z) &= (1.011) ~\frac{z-0.9756}{z-0.9975} \\ \\ &= (1.011)~\frac{1-0.9756 ~z^{-1}}{1-0.9975~z^{-1}} \\ \\H(e^{j\omega}) &= (1.011)~\frac{1-0.9756 ~e^{-j\omega}}{1-0.9975 ~e^{-j\omega}}\\ \end{align} $$

Then find the discrete-time radian frequency $\omega$ from the given continous-time frequency $\Omega = 3 ~\text{rad\s}$ by the sampling relation (assuming no aliasing):

$$ \Omega = 3 ~ \text{rad/s} \implies \omega = \Omega \times T_s = 3 \times 0.25 = \frac{3}{4} ~\text{rad/sample}$$

Then find the gain of the LTI system at the given frequency $\omega = 3/4$ :

$$ H(e^{j3/4}) = (1.011)\frac{1-0.9756 ~e^{-j3/4}}{1-0.9975 ~e^{-j3/4}} $$

Gain is the magnitude: $|H(e^{j3/4})| = 1.0004$. And the gain in dB is given by:

$$ G_{db} ~= 20 \log_{10}( |H(e^{j3/4})| ) ~= 0.0035 ~~\text{dB} $$

Fat32
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Note that the gain is obtained from the magnitude of the (complex-valued) transfer function evaluated at $z=e^{j\omega T}$, where $\omega$ is the angular frequency and $T$ is the sampling period.

Matt L.
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