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in my Course we talked about filter banks and the prof said that we get reduced resolution in time because of the downsampling after the band-pass filters. Is this because we now have fewer sample points per second to look at the narrowband signals?

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Anytime you take away "actual" data samples away by donwsampling, yes, you are loosing on finer granular data in time domain and hence reducing resolution. However, consider, the following case, an input signal is first up sampled by 2(interpolation) and then down sampled by 2, does it decrease resolution of actual data, not quite, because the upsampled new data points were anyways interpolated versions of the original data. So you have in total not lost any resolution of the actual data. So when we talk about resolution we should always be referring to resolution with reference to the actual signal data points

It's always helpful to keep in mind what happened before the down sampling.

Almost all the time in a system, a downsampling would be happening on an "oversampled signal"(for noise reduction purposes) or preceded by upsampling for rate conversion. In both cases, the data points that would be throwing off were in any case "not required" for processing.

In summary, yes, any time you throw away actual samples of the signal you do loose on information in time domain, but as illustrated above its important to keep in mind that what is being down sampled are actual or artificial data.

Side note on Frequency Domain: Any time we down-sample a signal, we are essentially limiting the signal bandwidth that can be represented unambiguously

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Yes, you are right. By downsampling we are reducing sample points per second (basically lowering the sampling rate). The time gap between adjacent samples increases as a result, hence degrading the resolution in time.

Imagine, your original sampling frequency was $\text{100kHz}$. Your time interval between samples was $\text{1/100k}$ = $\text{10us}$. If you down sample by $2$, the time difference between adjacent samples becomes $\text{20us}$. So the corresponding sampling rate is $\text{50kHz}$, which is half of the original rate.

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Down sampling usually requires filtering to prevent aliasing (unless the signal was already suitably bandlimited for the lower rate).

A low pass filter removes high frequency components. Part of the shape and location of signal transition edges in the time domain is represented by high frequency components in the frequency domain. Removing those high frequency components can not only distort the shape of a transition edge, but also move the exact location of zero-crossings, peaks, etc. Thus distorting or reducing time resolution of those events.

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