All signals in the real-world are continuous. In contrast, all signals stored in a computer use the unit of byte as the basic storage unit. Therefore these signals are referred to discrete-time signals. In mathematical notations, discrete-time signals can be represented by x(nT), where T is the sampling period (the reciprocal of the sample rate), n is an integer and nT is the elapsed time. For simplicity, we shall use x[n] to represent the discrete-time signal at time nT, with T implicitly defined.
In the following, we shall introduce some of the commonly used discrete-time signals:
The unit impulse signal d[n] is zero everywhere except at n = 0, where its value is 1. Mathematically, we can express the unit impulse signal as the following expression:
d[n] = 1, if n=0
d[n] = 0, otherwise
We can plot the unit impluse signal as follows:
If we delay the unit impulse signal by k sample points, we have the following equation:
d[n-k] = 1, if n=k
d[n-k] = 0, otherwise
The above delayed unit imulse signal can be plotted as follows:
The above delayed unit impulse signal have the property of masking out other signals not at n = k. In other words, if we multiply any signals x[n] (n = -∞∼∞) with d[n-k], only the term of x[n-k] is left. This can be expressed as the following equation:
x[k] = Sn=-∞∞d[n-k]•x[n]
By using this property, we can express any discrete-time signals as a linear combination of the unit impulse signals. This is especially handy when we derive the response of a linear time-invariant system. See the following section for more details.
The unit step signal u[n] is 1 when n≧0, and 0 elsewhere. In mathematical notations, we have
u[n] = 1, if n≧0
u[n] = 0, otherwise
The unit step signal can be plotted, as shown in the next example:
It should be noted that the unit impulse signal is the first-order difference of the unit step signal:
d[n] = u[n] - u[n-1].
The sinusoidal signal is another commonly used discrete-time signals:
sin[n] = sin(2pf(nT)) = sin(wn))
where f is the oscillation frequency of the sinusoidal signal and w ( = 2pfT) is the normalized angular frequency. The sinusoidal signal can be plotted as follows:
