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:

We can plot the unit impluse signal as follows:

d[n] = 1, if n=0

d[n] = 0, otherwise

If we delay the unit impulse signal by k sample points, we have the following equation:The above delayed unit imulse signal can be plotted as follows:

d[n-k] = 1, if n=k

d[n-k] = 0, otherwise

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 = -¡Û¡ã¡Û) with d[n-k], only the term of x[n-k] is left. This can be expressed as the following equation: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.

x[k] = S _{n=-¡Û}^{¡Û}d[n-k]¡Ex[n]The unit step signal u[n] is 1 when n¡Ù0, and 0 elsewhere. In mathematical notations, we have

The unit step signal can be plotted, as shown in the next example:

u[n] = 1, if n¡Ù0

u[n] = 0, otherwise

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 thenormalized angular frequency. The sinusoidal signal can be plotted as follows:

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