# What does filtering do to a signal in time domain? Does the signal gets distorted?

When considering a transceiver system, if we do a certain filtering operation on the signal, will the BER of the system be affected? My point is, since filtering in frequency domain filters a portion of the spectrum, whereas in time domain it is basically a series of MAC operations depending on the number of filter taps. So, eventually my original signal after filtering should have a totally different waveform than the original. Now, if I want to get the original signal back in an ideal channel condition, do we have to do some sort of inverse filtering operations that compensate the operations done in the time domain?

I will be grateful if you could help me in this regard. Thank you in advance!

• The use of linear phase FIR filter does not distort the shape of the waveform and preserves the waveshape. – DSP Novice Mar 17 at 10:55

It's always important to keep in mind that more than 80% of signal processing is filtering. However, not every filtering operation should be "viewed" in the same way.

Let us take the example you mention, of an "ideal known channel" (assuming it's not in deep fade). In wireless communications that would mean perfect knowledge of channel to the receiver. Typically, you could invert the channel in a very basic and often used scheme known as Zero forcing i.e. if $$y=hx + n$$ where y is received signal then the equalization is simply $$y/h$$ assuming real signals and noise. Similarly an MMSE equalizer ( optimal for most SNRs) has a typical mathematical expression that you would find in standard textbooks. The point is it would be less intuitive to look at this equalizer from the perspective of it's frequency response. It might be more intuitive to look at it from the perspective of vector spaces in this case. Especially as the degree of freedom of a network grow. For ex: a MIMO system.

Filtering is essential to iradicate BER, simply because if you don't filter ( as mentioned above), the time domain signal has been essentially "changed" by the channel.

We are always trying "mitigate" the effects of the channel. I would say "mitigate" rather than "invert" because it is sometimes not even possible or doesnt make sense to simply invert. For ex: in the scheme above $$y/h$$, notice that the noise is also devided by $$h$$. If $$h$$ is small, then you boost the noise as well. So in such a case a MMSE equalizer would make sense. When viewed in line of probability theory, the MMSE equalizer is calculating the expected value of the posterior distribution of $$x$$ (your signal) after observing $$y$$. So in that sense it is not inverting the channel.

Long story short, we try to do what's best for the system depending on bit rates, how close the bit rate is to channel capacity, SNR, modulation schemes etc. Rather than simply invert the channel.

There are other scenarios such as an antialsiaing filter, in such cases it is best viewed from it's frequency response, like cut off frqeuency and phase responses. You would get a better idea of the time domain filtered sequence looking at it from the frequency domain perspective. For ex: the cut off frqeuency would tell us what maximum gradients are tolerated in the input signal to the filter and hence are part of the filtered sequence or are smoothen out. The phase response would tell us which freqencies of the filtered sequence, would be delayed by what amount. Whether delay would be constant for the whole signal (linear phase) or there would be different group delays (non linear phase).

Indeed the filtering will change the shape of the input signal, otherwise usually nothing interesting is happening, but the filter itslef is not designed to "distort" the signal, it is designed to "better" the signal according to application, indeed the shape /magnitude of time domain filtered signal will change after filtering, but for the better of the system.

When considering a transceiver system, if we do a certain filtering operation on the signal, will the BER of the system be affected?

Yes it does. The whole of equalization concept in digital communication is to mitigate the effect of transmit signal getting filtered by channel (which includes right from the Transmit chain components, the communication medium like atmosphere, buildings, and receiver components).

Now, if I want to get the original signal back in an ideal channel condition, do we have to do some sort of inverse filtering operations that compensate the operations done in the time domain?

You need to explain more on what you meant by ideal channel. There are variety of channel models available for each of the commercial frequency bands. The signal under goes different form of filtering depending on the band. For example, in cellular and Wifi band (<6GHz), signal can penetrate walls, some get bounced due to non-absorption creating a multi-path effect. If you are in Line-Of-Sight you get a 'good' channel with strong reception. If you go to mmWave (>50GHz band) signal cannot penetrate walls and cannot travel more than 10 meters. So the equalization and training strategies are different depending on what technology/band you use. Equalization techniques like Minimum-Mean-Square-Estimate (MMSE), Adaptive Filtering to an extent can inverse the effect of channel.

• Nice answer but I think the mmWave statement isn't accurate. mmWave signals indeed cannot penetrate walls however they can go much further than 10 meters. The 60 GHz band in particular is most limited as it is in the oxygen absorption band but even than can be transmitted much much further than 10 meters with high gain antennas (here is an example where they reported reaching over 1 mile: ignitenet.com/technology/metrolinq). For other frequencies (such as E-band) the transmit range is much further, in many cases close to that as given by free-space path loss computation. – Dan Boschen Mar 17 at 23:48