# Filter response to sinusoidal input signal

This question is related to Does 46 dB gain of a filter for a frequency imply 200 times more amplitude? In the accepted answer it is mentioned that "For a sine wave input the amplitude of the steady state output is simply the amplitude of the input multiplied with the magnitude of the transfer function at that frequency"

Above relation is derived as an integral; as a convolution between filter and the input. But in practice we have all discrete signal, therefore can we get 200 times more in practice?

• no, the convolution theorem, and Parseval's, also hold true for discrete signals. Mar 1 '20 at 21:37
• @MarcusMüller I am not sure it is related to convolution theorem, it is related to the fact that there is no window function effect in integral and there is one in discrete case. Mar 2 '20 at 1:12
• it really is. The convolution theorem states exactly that the frequency response of a filter behaves like that: pointwise multiplication yields the amplitude at the given frequency. Mar 2 '20 at 9:44
• In essence, what you're asking is "does the definition of amplitude response still work?", and yes, it does. Mar 2 '20 at 9:45

The fact that a sinusoidal signal's amplitude is multiplied by the magnitude of the system's frequency response (evaluated at the input frequency) is a general property of LTI systems, regardless of whether they are continuous-time or discrete-time systems.

This is a consequence of the complex exponential being an eigenfunction of LTI systems, and the value of the frequency response evaluated at the input frequency is the corresponding eigenvalue.

For more details on eigenfunctions of LTI system check out this question and its answers. And I'm sure that there's even more information here on this site.

But in practice we have all discrete signal, therefore can we get 200 times more in practice?

Of course. Consider the simple system "multiply every sample with 200". That has a transfer function of $$H(\omega) = 200$$ and you get 200 times the input for any input signal.

Let's look at a more practical example: a typical audio high pass filter. Let's say it's it is a 2nd order butterworth with a cutoff of 40 Hz at sample rate of 44.1kHz implemented as a Direct Form II filter. While the overall amplitude gain of the filter never exceeds 0 dB, the transfer function from the input to the internal state variable is given by the pole-only transfer function. This has an amplitude gain at low frequencies of almost 90 dB (> 30000).

It takes a few samples for the output to build up, but it goes up fairly quickly. If you feed the filter a DC signal, you will reach a gain of 200 after about 20 samples. So these types of gain or very real and can happen in practice all the time.