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I guess the most straightforward way of adding an artificial echo to a sound signal is to add its time-delayed version to the original signal. However, I am not sure how realistic that would sound like.

I was reading papers on room geometry estimation from impulse responses (see for example this paper). So, perhaps one should have several echoes instead of just one.

I guess something also happens to a sound signal when it is reflected (kind of filtering).

What would you suggest here?

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A very simple model is a sum of attenuated and delayed replicas of the signal. Such signal can be be generated using FIR filter. The length of the filter is the "echo time", i.e. how long the echoes last.
The simplest approach would be to generate sparse impulse response. Build a filter with $n$ non-zero coefficient and fading amplitude (for example the amplitude can be generated with Rayleigh distribution).

If you want a more realistic scenario, I suggest using the Room Impulse Response (RIR) generator by Emanuel Habets.

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There are two different things you could be talking a out: delay and reverb. I think what you are looking for is reverb.

Two ways to do reverb are multitap reverb and convolution reverb. Multitap is faster but convolution is higher quality.

Here are some links with more info and example c++ code:

http://blog.demofox.org/2015/03/17/diy-synth-multitap-reverb/

http://blog.demofox.org/2015/03/23/diy-synth-convolution-reverb-1d-discrete-convolution-of-audio-samples/

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