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I know that when a signal has finite energy,it is energy signal and when its power is finite, it is power signal

But what is exactly meaning of finite here? Zero also included?

For example if a signal has zero energy, will its be considered energy signal? Similarly if its power is zero, will it be considered power signal?

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There's only one signal with zero energy, namely the signal $x(t)=0$, which is not a very interesting signal. All energy signals, i.e., all signals with finite energy, have zero power. That's why it is pointless to refer to them as "power signals". So "power signals" are all signals with finite - but non-zero - power.

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"Finite" indeed may cause confusion. We also use it as in "a finite set of values": $\{2,7,13,17\}$. From a mathematical point-of-view, finite means "a non infinite scalar value", and zero would be allowed. Bounded periodic functions like sines are typically with "infinite energy": they are not $L_2$-norm integrable, but they have finite power. A function with zero-energy could be not-zero (on a set of null-measure), but treated equivalently as the null function, as in a class of equivalence.

In DSP though, it is traditional to rule out zero-energy and zero-power signals: an energy signal has zero average power, and a power signal has infinite energy. There are more details here: for the continuous How to check if a signal is power signal or energy signal? and the discrete case Power or Energy Signal?

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