# 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?

short answer: you can (approximately) implement a lowpass filter by sampling its prototype continuous-time impulse response $$h_c(t)$$, but you cannot implement an ideal highpass filter that way, since the highpass filter is non-bandlimited; i.e., there will be unavoidable aliasing errors due to the definition of the highpass filter, but read on...

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

This operation is analogous to sampling, and 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 below the Nyquist rate associated with $$h_c(t)$$.

A continous-time ideal lowpass filter is by definition bandlimited (or a practical one will be sufficiently bandlimited), and we can always find a small enough $$T_d$$ to sample its impulse response while avoiding aliasing.

On the other hand, an ideal continuous-time highpass filter has an infinite bandwidth, and thus sampling period required is $$T_s = 0$$; i.e., it cannot be sampled. The technique is not applicable to such a system.

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 you can replace the ideal highpass filter by an ideal bandpass one, (or one of its practical approximations). When such is the case, then you can apply the technique of impulse invariance to obtain a discrete-time band pass filter from its continuous-time prototype, by choosing a suitable sampling period $$T_s$$ to completely avoid (or practically minimize) the resulting aliasing errors...

Note that the method of impulse invariance produces an exact linear mapping from continuous-time to discrete-time frequency spectrum. It preserves all characteristics of the continuous-time filter, unlike the method of bilinear transform which warps the spectrum while mapping it.

• 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. Commented Jul 9, 2016 at 21:07
• yes that's right... I think I've mixed ideal filters with the practical ones... Commented Jul 9, 2016 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 Commented Jul 10, 2016 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. Commented Jul 10, 2016 at 10:21