# Sampling the impulse response

Given two IIR filters (one Low-Pass and one High-Pass) designed using Butterworth, how can I know if they can be obtained using sampling the impulse response?

The short answer is: You can implement a lowpass filter by sampling its prototype continuous time impulse response $h_c(t)$, but you cannot implement a highpass filter by sampling its prototype impulse response, due to highpass filter being non-bandlimited i.e. there will be unavoidable aliasing due to the definition of the highpass filter. But read on.

The technique of obtaning a discrete time filter impulse response $h_d[n]$ from a continuous time prototype filter by sampling its impulse response $h_c(t)$ as given by $$h_d[n] = T_s h_c(nT_s)$$ is called as the method of impulse-invariance , where $T_s$ is the sampling period.

This operation of obtaning a discrete time filter from a continuous time prototype is analogous to the more traditional operation of signal sampling. In fact their implications are the same: There will be aliasing at the resulting discerete time spectrum of the converted filter if either 1-The continuous time filter is not bandlimited or 2-The sampling rate is not adequate, given by the Nyquist rate as applied to sampling of bandlimited signals.

A continous time lowpass filter by definition, is always bandlimited, and we can always, therefore, find a small enough $T_d$ which will avoid aliasing at the resulting discerete time spectrum of the filter being converted.

On the other hand, a continuous time highpass filter is going to a have an infinite bandwidth and as such the sampling period required is $T_s = 0$ , i.e. it cannot be sampled.

However, if you will be carefull enough, and if you can be sure about the bandwidth requirements of every stage in your signal processing chain, then may be you can replace a highpass filter by a bandpass one, when such is the case, then you can still apply the technique of impulse invariance to obtain a discrete time band pass filter from a continuous time bandpass filter prototype, by choosing a suitable sampling period $T_s$

Note that, the method of impulse invariance produces an exact linear mapping from continuous time to discrete time frequency spectrum. It will preserve all the characteristics of the continuous time filter (unlike the method of bilinear transform which warps the spectrum)

• Just to avoid confusion: a (causal and stable) continuous-time low pass filter is never band-limited. It's just that we can always find a sampling interval $T$ such that the aliasing error becomes acceptable. However, there will always be aliasing, also for low pass filters. – Matt L. Jul 9 '16 at 21:07
• yes that's right... I think I've mixed ideal filters with the practical ones... – Fat32 Jul 9 '16 at 22:33
• That's true Matt. But, for a low-pass filter it's good to consider that the information that I lose saying that it's band-limited is insignificant – Euler Jul 10 '16 at 0:53
• @Euler: That's exactly the question: is the error insignificant? You can't just say that it is, you need to check it depending on the application, and there are many cases where you can't just ignore the aliasing error. That's especially true if the cut-off frequency is relatively close to Nyquist. – Matt L. Jul 10 '16 at 10:21