# Electrical Signal ## Size of a Signal Size or signal strength? Such a measure must consider not only the signal amplitude but also its duration. For instance, if we are to devise a single number $ V $ as a measure of the size of a human being, we must consider not only his or her width (girth), but also the height. The product of girth and height is a reasonable measure of the size of a person. If we wish to be a little more precise, we should average this product over the entire length of the person. If we make the simplifying assumption that the shape of a person is a cylinder of radius $ r $, which varies with the height $ h $ of the person, then a reasonable measure of the size of a person of height $ H $ is the person’s volume $ V $, given by $$ V = \pi \int_{0}^{H} r^2(h) \, dh $$ ## Signal Energy Arguing in this manner, we may consider the area under a signal $ g(t) $ as a possible measure of its size, because it takes account of not only the amplitude but also the duration. However, this will be a defective measure because $ g(t) $ could be a large signal, yet its positive and negative areas could cancel each other, indicating a signal of small size. This difficulty can be corrected by defining the signal size as the area under $ g^2(t) $, which is always positive. We call this measure the **signal energy** $ E_g $, defined (for a real signal) as $$ E_g = \int_{-\infty}^{\infty} g^2(t) \, dt $$ This definition can be generalized to a complex-valued signal $ g(t) $ as $$ E_g = \int_{-\infty}^{\infty} |g(t)|^2 \, dt $$ There are also other possible measures of signal size, such as the area under $ |g(t)| $. The above energy measure, however, is not only more tractable mathematically but is also more meaningful (as shown later) in the sense that it is indicative of the energy that can be extracted from the signal. ## Signal Power The signal energy must be finite for it to be a meaningful measure of the signal size. A necessary condition for the energy to be finite is that the signal amplitude $ \to 0 $ as $ |t| \to \infty $. Otherwise, the integral in Eq. (2.1) will not converge. If the amplitude of $ g(t) $ does not $ \to 0 $ as $ |t| \to \infty $, the signal energy is infinite. A more meaningful measure of the signal size in such a case would be the time average of the energy (if it exists), which is the **average power** $ P_g $ defined (for a real signal) by $$ P_g = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} g^2(t) \, dt $$ We can generalize this definition for a complex signal $ g(t) $ as $$ P_g = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} |g(t)|^2 \, dt $$ Observe that the **signal power** $ P_g $ is the time average (mean) of the signal amplitude squared, that is the **mean-squared** value of $ g(t) $. Indeed, the square root of $ P_g $ is the familiar **root mean square (rms)** value of $ g(t) $.