- 1.1 Discrete Sequences and Their Notation
- 1.2 Signal Amplitude, Magnitude, Power
- 1.3 Signal Processing Operational Symbols
- 1.4 Introduction to Discrete Linear Time-Invariant Systems
- 1.5 Discrete Linear Systems
- 1.6 Time-Invariant Systems
- 1.7 The Commutative Property of Linear Time-Invariant Systems
- 1.8 Analyzing Linear Time-Invariant Systems
- References
- Chapter 1 Problems

## 1.5 Discrete Linear Systems

The term *linear* defines a special class of systems where the output is the superposition, or sum, of the individual outputs had the individual inputs been applied separately to the system. For example, we can say that the application of an input *x*_{1}(*n*) to a system results in an output *y*_{1}(*n*). We symbolize this situation with the following expression:

#### Equation 1–11

Given a different input *x*_{2}(*n*), the system has a *y*_{2}(*n*) output as

#### Equation 1–12

For the system to be linear, when its input is the sum *x*_{1}(*n*) + x_{2}(*n*)*,* its output must be the sum of the individual outputs so that

#### Equation 1–13

One way to paraphrase expression (1–13) is to state that a linear system's output is the sum of the outputs of its parts. Also, part of this description of linearity is a proportionality characteristic. This means that if the inputs are scaled by constant factors *c*_{1} and *c*_{2}, then the output sequence parts are also scaled by those factors as

#### Equation 1–14

In the literature, this proportionality attribute of linear systems in expression (1–14) is sometimes called the *homogeneity property*. With these thoughts in mind, then, let's demonstrate the concept of system linearity.

### 1.5.1 Example of a Linear System

To illustrate system linearity, let's say we have the discrete system shown in Figure 1–7(a) whose output is defined as

#### Equation 1–15

Figure 1–7 Linear system input-to-output relationships: (a) system block diagram where y(n) = –x(n)/2; (b) system input and output with a 1 Hz sinewave applied; (c) with a 3 Hz sinewave applied; (d) with the sum of 1 Hz and 3 Hz sinewaves applied.

that is, the output sequence is equal to the negative of the input sequence with the amplitude reduced by a factor of two. If we apply an *x*_{1}(*n*) input sequence representing a 1 Hz sinewave sampled at a rate of 32 samples per cycle, we'll have a *y*_{1}(*n*) output as shown in the center of Figure 1–7(b). The frequency-domain spectral amplitude of the *y*_{1}(*n*) output is the plot on the right side of Figure 1–7(b), indicating that the output comprises a single tone of peak amplitude equal to –0.5 whose frequency is 1 Hz. Next, applying an *x*_{2}(*n*) input sequence representing a 3 Hz sinewave, the system provides a *y*_{2}(*n*) output sequence, as shown in the center of Figure 1–7(c). The spectrum of the *y*_{2}(*n*) output, *Y*_{2}(*m*), confirming a single 3 Hz sinewave output is shown on the right side of Figure 1–7(c). Finally—here's where the linearity comes in—if we apply an *x*_{3}(*n*) input sequence that's the sum of a 1 Hz sinewave and a 3 Hz sinewave, the *y*_{3}(*n*) output is as shown in the center of Figure 1–7(d). Notice how *y*_{3}(*n*) is the sample-for-sample sum of *y*_{1}(*n*) and *y*_{2}(*n*). Figure 1–7(d) also shows that the output spectrum *Y*_{3}(*m*) is the sum of *Y*_{1}(*m*) and *Y*_{2}(*m*). That's linearity.

### 1.5.2 Example of a Nonlinear System

It's easy to demonstrate how a nonlinear system yields an output that is not equal to the sum of *y*_{1}(*n*) and *y*_{2}(*n*) when its input is *x*_{1}(*n*) + *x*_{2}(*n*). A simple example of a nonlinear discrete system is that in Figure 1–8(a) where the output is the square of the input described by

#### Equation 1–16

Figure 1–8 Nonlinear system input-to-output relationships: (a) system block diagram where y(n) = [x(n)/2]; (b) system input and output with a 1 Hz sinewave applied; (c) with a 3 Hz sinewave applied; (d) with the sum of 1 Hz and 3 Hz sinewaves applied.

We'll use a well-known trigonometric identity and a little algebra to predict the output of this nonlinear system when the input comprises simple sinewaves. Following the form of Eq. (1–3), let's describe a sinusoidal sequence, whose frequency *f*_{o} = 1 Hz, by

#### Equation 1–17

Equation (1–17) describes the *x*_{1}(*n*) sequence on the left side of Figure 1–8(b). Given this *x*_{1}(*n*) input sequence, the *y*_{1}(*n*) output of the nonlinear system is the square of a 1 Hz sinewave, or

#### Equation 1–18

We can simplify our expression for *y*_{1}(*n*) in Eq. (1–18) by using the following trigonometric identity:

#### Equation 1–19

Using Eq. (1–19), we can express *y*_{1}(*n*) as

#### Equation 1–20

which is shown as the all-positive sequence in the center of Figure 1–8(b). Because Eq. (1–19) results in a frequency sum (a + b) and frequency difference (a – b) effect when multiplying two sinusoids, the *y*_{1}(*n*) output sequence will be a cosine wave of 2 Hz and a peak amplitude of –0.5, added to a constant value of 1/2. The constant value of 1/2 in Eq. (1–20) is interpreted as a zero Hz frequency component, as shown in the *Y*_{1}(*m*) spectrum in Figure 1–8(b). We could go through the same algebraic exercise to determine that when a 3 Hz sinewave *x*_{2}(*n*) sequence is applied to this nonlinear system, the output *y*_{2}(*n*) would contain a zero Hz component and a 6 Hz component, as shown in Figure 1–8(c)

System nonlinearity is evident if we apply an *x*_{3}(*n*) sequence comprising the sum of a 1 Hz and a 3 Hz sinewave as shown in Figure 1–8(d). We can predict the frequency content of the *y*_{3}(*n*) output sequence by using the algebraic relationship

#### Equation 1–21

where *a* and *b* represent the 1 Hz and 3 Hz sinewaves, respectively. From Eq. (1–19), the *a*
^{2} term in Eq. (1–21) generates the zero Hz and 2 Hz output sinusoids in Figure 1–8(b). Likewise, the *b*
^{2} term produces in *y*_{3}(*n*) another zero Hz and the 6 Hz sinusoid in Figure 1–8(c). However, the 2*ab* term yields additional 2 Hz and 4 Hz sinusoids in *y*_{3}(*n*). We can show this algebraically by using Eq. (1–19) and expressing the 2*ab* term in Eq. (1–21) as

#### Equation 1–22

^{†}

Equation (1–22) tells us that two additional sinusoidal components will be present in *y*_{3}(*n*) because of the system's nonlinearity, a 2 Hz cosine wave whose amplitude is +1 and a 4 Hz cosine wave having an amplitude of –1. These spectral components are illustrated in *Y*_{3}(*m*) on the right side of Figure 1–8(d).

Notice that when the sum of the two sinewaves is applied to the nonlinear system, the output contained sinusoids, Eq. (1–22), that were not present in either of the outputs when the individual sinewaves alone were applied. Those extra sinusoids were generated by an interaction of the two input sinusoids due to the squaring operation. That's nonlinearity; expression (1–13) was not satisfied. (Electrical engineers recognize this effect of internally generated sinusoids as *intermodulation distortion*.) Although nonlinear systems are usually difficult to analyze, they are occasionally used in practice. References [2], [3], and [4], for example, describe their application in nonlinear digital filters. Again, expressions (1–13) and (1–14) state that a linear system's output resulting from a sum of individual inputs is the superposition (sum) of the individual outputs. They also stipulate that the output sequence *y*_{1}(*n*) depends only on *x*_{1}(*n*) combined with the system characteristics, and not on the other input *x*_{2}(*n*); i.e., there's no interaction between inputs *x*_{1}(*n*) and *x*_{2}(*n*) at the output of a linear system.