Clarify the Difference between the Filter and Output Response

[Ahmed Abdellah]

I am confused between the terminology of the frequency representation of the filter and the magnitude spectrum of the output.

So I thought that the former is called “frequency representation of the filter ” and the latter is called “frequency response”

but at the video : Source – filter model, at 9:30, Prof Simon called the former as “frequency response” !

Is that right? I supposed that the term “response” is the same as “output”

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[Catherine Lai]

The frequency response of a filter describes how a filter interacts with input signals in the the frequency domain. We can then talk about the frequency magnitude response as a curve of frequencies (ranging from 0 Hz to the Nyquist frequency) versus magnitudes.

So, you can think of the frequency response of a filter telling you how the applying a filter to an input signal would change the frequency spectrum of the input signal (aka the frequency response of the DFT to the input!). For example, you might design a low pass filter that completely attenuates all frequency components that are greater that 8000 Hz. In theory this filter would have a magnitude response curve that goes to zero for all frequency values above 8000 Hz.

To see that the frequency response of a filter is something different (but closely related) to the magnitude spectrum of a specific waveform, you can think about impulse trains with different fundamental frequencies.

We know that impulse trains with different fundamental frequencies have different harmonics, so their magnitude spectra differ in terms of which frequencies would get non-zero magnitudes. For example, if we had an impulse train with F0=100Hz, only frequencies which were multiples of 100Hz would have positive magnitudes, but if our impulse train had F0=200Hz, only multiples of 200Hz would be positive. However, if we applied the same filter to both impulse trains, both of their magnitude spectra would have the same overall shape (i.e. spectral envelope), which would match the shape of the filter’s frequency magnitude response! (You can use the code in the Module 2 FIR and IRR filter notebooks to check this!)

So, to get down to the terminological question ‘the frequency representation of the filter’ is the same as the ‘frequency response’ of the filter. This determines what the filter will do to the frequency components of the input signal. If we do the DFT of the output of applying the filter to an input in the time domain it will have the same overall shape of the frequency response of the filter, but the actual details of the magnitude spectrum will depend on the frequency components of the input signal.

Side note: it’s good to remember that there are actually two parts of the frequency spectrum: the magnitude spectrum and the phase spectrum. So the frequency response of a filter also includes a separate phase response. Most of the time, for speech technology applications, when we talk about the frequency response of a filter, we’re just talking about the frequency magnitude response but it’s worth noting that a filter can have an effect of phase shift too (see the moving average example in the FIR filter notebook!).

[Isobel W]

So the magnitude spectrum (a.k.a spectral envelope) is about an overall pattern, ratios and scales. In contrast, the frequency response is about the components (different frequencies) involved. Might not work as an analogy but I was thinking it’s akin to two buildings. They both might have the same tower block structure, but one may be made of concrete, the other made of brick. To bring it back to the impulse train – ALL impulse trains have the same structure – periodic with frequencies at every harmonic – however these frequencies can be different (multiples of 100Hz vs 200Hz) – but the overall periodic pattern is the same

[Simon]

The magnitude spectrum is not exactly the same thing as the spectral envelope.

The magnitude spectrum is something that can be obtained for any signal, such as speech, by taking the DFT of the signal (and discarding phase).

Looking at the magnitude spectrum of speech, we see two very different properties combined. One is the overall shape – the spectral envelope. The other is the fine structure.

For speech, the frequency response of the vocal tract filter is responsible for creating the spectral envelope of the speech signal.

For voiced speech, the fine structure will be a regular series of harmonics at multiples of F0, and this is created by the vocal folds.

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