# Can we band-limit a PCM signal? Is it even relevant to relate a totally Analog term(Band limit) with digital one(PCM)

Can we band-limit a PCM signal?

Yes -- it's just a matter of selecting the right pulse shape.

(By the way, it makes perfect sense to talk about the bandwidth of a PCM signal; it is, after all, a continuous-time physical signal with bandwidth, energy, power, etc, just like any other signal.)

Most introductory textbooks (irritatingly) assume that a PCM signal must use rectangular pulses. You don't have to, and actually you almost never do, except in the simplest applications.

The PCM signal is a train (or sequence) of pulses with varying amplitudes. This train can be written as $$s_{\text{PCM}}(t) = \sum_k a_k p(t-kT_p),$$ where $$T_p$$ is the pulse rate, and $$a_k$$ is the amplitude of each pulse.

In the simplest PCM signal, $$a_k$$ are either 0 or 1, and $$p(t)$$ is a rectangular pulse of duration $$T_p$$. The bandwidth of $$s_{\text{PCM}}(t)$$ is the same as the bandwidth of $$p(t)$$, which in this case is (mathematically) infinite.

By selecting a different pulse $$p(t)$$, you can limit the bandwidth of $$s_{\text{PCM}}(t)$$. A popular pulse shape is the "raised cosine", which has a bandwidth between $$1/T_p$$ and $$1/2T_p$$.

Note that the bandwidth can also be limited by "brute force": just filter the rectangular $$s_{\text{PCM}}(t)$$ with a low-pass filter of the desired cutoff frequency. This will distort the signal, but in simple applications (high SNR, no further distortion or attenuation) this usually causes no significant performance degradation.