Gain from Difference Equation (Revised)

Calculus Level 3

Consider a discrete infinite-impulse-response (IIR) filter, described by the following difference equation:

$\large{y_k = a \, x_k + b \, x_{k-1} + c \, y_{k-1}}$ In the above equation, $y$ is the filter output signal and $x$ is the filter input signal. The $(k)$ subscript denotes a value from the present processing interval, and the $(k-1)$ subscript denotes a value from the previous processing interval.

The filter coefficients are $(a,b,c) = (3,2,\frac{1}{2})$ .

What is the magnitude of the filter transfer function (ratio of output sinusoid magnitude to input sinusoid magnitude), assuming that the filter is processed at a rate of $1 \, \text{kHz}$ , and that the input signal frequency is $60 \, \text{Hz}$ ? This could also be referred to as the filter gain.

Note: I have revised this problem since the first posting. The original filter coefficients caused the filter to be unstable.

The answer is 8.68556.

1 solution

Steven Chase
Jan 12, 2018

There are two strategies for solving this problem. Both yield the same result:

1) Apply z-transforms and complex substitution for z
2) Simulate the difference equation to verify the gain

Z-transform method:

Difference equation:

$\large{y_k = a \, x_k + b \, x_{k-1} + c \, y_{k-1}}$

Applying the z-transform and using the time-shift property:

$Y = a \, X + b z^{-1} X + c \, z^{-1} Y$

Consolidating $Y$ and $X$ terms and forming the transfer function:

$Y (1 - c \, z^{-1}) = X (a + b z^{-1}) \\ \frac{Y}{X} = \frac{(a + b z^{-1})}{(1 - c \, z^{-1})}$

Subsitution for z:

$z = e^{j \omega T} \\ j = \sqrt{-1} \\ \omega = 2 \pi f = 2 \pi (60) \\ T = \frac{1}{1000}$

Plugging in numbers, we get a complex number which contains both the magnitude and phase responses. The magnitude of the transfer function (gain) is equal to 8.68556.

Simulation method:

The simulation is easy to perform using a computer. Applying a unit-magnitude sinusoid yields an output sinusoid with the same magnitude calculated in the formal solution.

Gain from Difference Equation (Revised)

The answer is 8.68556.

1 solution

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