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# Energy and Power of Discrete-Time Signals

Module by: Marco F. Duarte. E-mail the author

Summary: Energy and power for discrete time signals

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## Signal Energy

### Discrete signals

For time discrete signals the "area under the squared signal" makes no sense, so we will have to use another energy definiton. We define energy as the sum of the squared magnitude of the samples. Mathematically

#### Energy - Discrete time signal:

Ed=n=|xn|2 Ed n x n 2

#### Example 1

Given the sequence yl=blul y l b l u l , where u(l) is the unit step function. Find the energy of the sequence.

We recognize y(l) as a geometric series. Thus we can use the formula for the sum of a geometric series and we obtain the energy, Ed=l=0yl2=11b2 Ed l 0 y l 2 1 1 b 2 . This expression is only valid for |b|<1 b 1 . If we have a larger |b|, the series will diverge. The signal y(l) then has infinite energy. So let's have a look at power...

## Signal Power

Our definition of energy seems reasonable, and it is. However, what if the signal does not decay fast enough? In this case we have infinite energy for any such signal. Does this mean that a fifty hertz sine wave feeding into your headphones is as strong as the fifty hertz sine wave coming out of your outlet? Obviously not. This is what leads us to the idea of signal power, which in such cases is a more adequate description.

### Discrete signals

For time discrete signals we define power as energy per sample.

#### Power - Discrete time:

Pd=1Nn=N1N1+N1|xn|2 Pd 1 N n N1 N1 N 1 x n 2

#### Example 2

Given the signals x1t=sin2πt x1 t 2 t and x2n=sinπ110n x2 n 1 10 n , shown in Figure 2, calculate the power for one period.

For the analog sine we have Pa=1101sin22πtdt=12 Pa 1 1 t 0 1 2 t 2 1 2 .

For the discrete sine we get Pd=120n=120sin2110πn=0.500 Pd 1 20 n 1 20 1 10 n 2 0.500 . Download power_sine.m for plots and calculation.

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