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I'm currently solving the following convolution problem from Oppenheimer's book: enter image description here

In the solution, it was stated that "$x(t)$ periodic implies $y(t)$ is periodic" So I wondered if it's always the case that if $x(t)$ is periodic then so is the output from the convolution, I tried doing research about this in hope of finding any supporting evidence to this but I couldn't, and "circular convolution" is all around the place.

Essam
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  • See https://dsp.stackexchange.com/questions/19799/periodicity-of-the-output-of-causal-time-invariant-systems-when-the-input-is-per for a previous discussion about this topic. – MBaz Apr 17 '21 at 22:23

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I am somehow rephrasinge what @Hilmar already answered. In the continuous-time setting, convolution cannot invent frequencies. Indeed, there is an intimate relationship between Fourier and convolution, that is not always taught properly. Recently, Michael Bronstein was Deriving convolution from first principles

Have you ever wondered what is so special about convolution? In this post, I derive the convolution from first principles and show that it naturally emerges from translational symmetry.

The basic assumptions that a system is linear, and invariant by shift or in time (LTI) imply the concept of convolution. And convolution implies that a sine undergoing an LTI system will still be a sine, albeit with a time-shift (phase change) and amplitude modification (it can even vanish). In other symbols, through an LTI:

$$ \sin(\omega t)\to A\sin(\omega t+\phi)$$

$A$ can be zero, so frequencies can disappear. Yet if $\sin(\omega t)$ is not present in the original data (no $\omega$), it cannot reappear. Frequency creation from nothing is a feature of nonlinear or time-variant systems.

Laurent Duval
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    Nitpick: frequency creation is a feature of time-variant systems. For example, the system $\text{T}[x(t)] = x(t)\cos(\omega t)$ is linear but it can create new frequencies not present in $x(t)$. – MBaz Apr 18 '21 at 15:17
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    Interesting post by Bronstein BTW, thanks for linking to it! – MBaz Apr 18 '21 at 15:20
  • Of course, I was in the state of mind on shift-invariance. Correcting – Laurent Duval Apr 18 '21 at 17:00
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So I wondered if it's always the case that if x(t) is periodic then so is the output from the convolution

Yes, it's always the case. Convolution is a linear operation and linear operations preserve periodicity.

Roughly speaking: any periodic signal consists of a sum of discrete sine waves. The output of an LTI system to a sine wave is also a sine wave at the same frequency. Hence the LTI system preserves the "discreteness" of the original input.

Hilmar
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  • Alright, is it then safe to say that the output signal will also have the same period? Because I think I also need that fact. – Essam Apr 17 '21 at 21:31
  • Yes. LTI systems can't shift frequencies around. The complex exponential is an eigenfunction of an LTI system. – MackTuesday Apr 17 '21 at 22:42
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    No. You can construct examples where the output has half, third, fourth .. the period of the input signal. @MackTuesday: LTI can't shift frequencies but it can eliminate specific ones. – Hilmar Apr 17 '21 at 23:19
  • Of course. I knew I was missing something. – MackTuesday Apr 18 '21 at 00:11
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    You picked the wrong one to single out. It's not the linearity that preserves periodicity, it's time-invariance. Linearity is neither sufficient nor necessary. – Jazzmaniac Apr 18 '21 at 11:57