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In a multipath propagation environment if we take the time index of reach channel tap by the integer multiples of sampling periods, assume the RMS delay spread $ \sigma_{tau} $ and maximum excess delay $ \tau _m $ is known, then how to determine the maximum number of paths?

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You can't really tell that from anything. The maximum delay spread together with your sampling rate, however, gives you the number of paths that you can tell apart with a single observation of the channel. If you model every sample of your discrete-time channel impulse response as a path.

That's not really a sensible "channel model", however: the tapped delay line model works well enough for point reflectors and static channels. However, your channel changes, and your reflectors are moving: you get paths as "smeared" peaks in your ambiguity function.

Still, the tapped delay line model with each sample of the channel impulse response being one tap is immensely useful, as it allows you to equalize the channel by OFDM.

Many scenarious consider point reflectors (which sadly don't always sit "exactly" sampling periods apart) that are moving, thus "smearing" and leaking across multiple taps. As soon as you add anything that starts to track different path components, you might be able to sense more paths than you have taps!

So, honestly, there's no "maximum number of paths" that you can distinguish if you have enough time and if they are uncorrelated enough.

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