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1.8 Analyzing Linear Time-Invariant Systems

As previously stated, LTI systems can be analyzed to predict their performance. Specifically, if we know the unit impulse response of an LTI system, we can calculate everything there is to know about the system; that is, the system's unit impulse response completely characterizes the system. By "unit impulse response" we mean the system's time-domain output sequence when the input is a single unity-valued sample (unit impulse) preceded and followed by zero-valued samples as shown in Figure 1–11(b).

Figure 1-11

Figure 1–11 LTI system unit impulse response sequences: (a) system block diagram; (b) impulse input sequence x(n) and impulse response output sequence y(n).

Knowing the (unit) impulse response of an LTI system, we can determine the system's output sequence for any input sequence because the output is equal to the convolution of the input sequence and the system's impulse response. Moreover, given an LTI system's time-domain impulse response, we can find the system's frequency response by taking the Fourier transform in the form of a discrete Fourier transform of that impulse response[5]. The concepts in the two previous sentences are among the most important principles in all of digital signal processing!

Don't be alarmed if you're not exactly sure what is meant by convolution, frequency response, or the discrete Fourier transform. We'll introduce these subjects and define them slowly and carefully as we need them in later chapters. The point to keep in mind here is that LTI systems can be designed and analyzed using a number of straightforward and powerful analysis techniques. These techniques will become tools that we'll add to our signal processing toolboxes as we journey through the subject of digital signal processing.

In the testing (analyzing) of continuous linear systems, engineers often use a narrow-in-time impulsive signal as an input signal to their systems. Mechanical engineers give their systems a little whack with a hammer, and electrical engineers working with analog-voltage systems generate a very narrow voltage spike as an impulsive input. Audio engineers, who need an impulsive acoustic test signal, sometimes generate an audio impulse by firing a starter pistol.

In the world of DSP, an impulse sequence called a unit impulse takes the form

Equation 1–26


The value A is often set equal to one. The leading sequence of zero-valued samples, before the A-valued sample, must be a bit longer than the length of the transient response of the system under test in order to initialize the system to its zero state. The trailing sequence of zero-valued samples, following the A-valued sample, must be a bit longer than the transient response of the system under test in order to capture the system's entire y(n) impulse response output sequence.

Let's further explore this notion of impulse response testing to determine the frequency response of a discrete system (and take an opportunity to start using the operational symbols introduced in Section 1.3). Consider the block diagram of a 4-point moving averager shown in Figure 1–12(a). As the x(n) input samples march their way through the system, at each time index n four successive input samples are averaged to compute a single y(n) output. As we'll learn in subsequent chapters, a moving averager behaves like a digital lowpass filter. However, we can quickly illustrate that fact now.

Figure 1-12

Figure 1–12 Analyzing a moving averager: (a) averager block diagram; (b) impulse input and impulse response; (c) averager frequency magnitude response.

If we apply an impulse input sequence to the system, we'll obtain its y(n) impulse response output shown in Figure 1–12(b). The y(n) output is computed using the following difference equation:

Equation 1–27


If we then perform a discrete Fourier transform (a process we cover in much detail in Chapter 3) on y(n), we obtain the Y(m) frequency-domain information, allowing us to plot the frequency magnitude response of the 4-point moving averager as shown in Figure 1–12(c). So we see that a moving averager indeed has the characteristic of a lowpass filter. That is, the averager attenuates (reduces the amplitude of) high-frequency signal content applied to its input.

OK, this concludes our brief introduction to discrete sequences and systems. In later chapters we'll learn the details of discrete Fourier transforms, discrete system impulse responses, and digital filters.

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