# Why does time-domain convolution correspond to frequency-domain multiplication? (visual)

Signal Processing Asked on November 27, 2020

I seek a visual explanation of this. I’ve already seen the maths, and can derive the proofs – they amount to nill for an intuitive understanding. Any amount of math is welcome, as long as serving to ultimately explain it visually.

Examples of excellent explanations:

Continuous vs Discrete: both are welcome, but ultimately the discrete case must be explainable. I’ve long thought of the input-side algorithm myself for continuous convolution, but never completed the picture; real analysis is tricky.

Circular vs linear: a complete explanation ought to cover both, but I’m primarily interested in linear convolution (rather, how the circular of padded signals is equivalent to linear).

Duality: ideally, should be covered (conv in freq domain <=> mult in time domain).

Ideas:

1. Only right-padding works, and both inputs are padded; this looks like forcing inputs to correlate with lower and fractional frequencies (and more frequencies) relative to unpadded’s frame.
2. Something about convolving with shifted deltas and observing modulations of complex sinusoids in other domain; awaiting clarification from @AndyWalls.

## Related Questions

### Terminology for instantaneous phase of magnitude component of complex signal

1  Asked on November 4, 2020

### Design discrete controller for zero steady state error

2  Asked on October 26, 2020 by john-katsantas

### Complex Samples (IQ) – Baseband Filtering

2  Asked on October 24, 2020 by tiaro

### Does any paper use energy harvest to in the baseband?

1  Asked on October 16, 2020 by care-we

### Image Noise Detection Using MATLAB

2  Asked on October 15, 2020 by premnath-d

### Linear prediction (LPC) of Sine wave samples around maximas

1  Asked on October 5, 2020 by skaveelicious

### What is the point of smoothing an FFT or spectral density plot, and how does that affect the noise floor?

2  Asked on September 27, 2020 by user3308243

### Interpreting the cross correlation results

1  Asked on September 7, 2020 by sampath

### How can I convert audio to a series of notes using pitch detection?

3  Asked on August 14, 2020 by nagasgura

### Matlab CIC decimator output mirrored

0  Asked on August 7, 2020 by edson-manoel