Chapter 17

Ten Properties You Never Want to Forget

In This Chapter

Looking at significant properties you can’t live without as a signals and systems engineer

Seeing properties side by side between continuous and discrete signals and systems

A big wide world of properties is associated with signals and systems — plenty in the math alone! In this chapter, I present ten of my all-time favorite properties related to signals and systems work.

LTI System Stability

Linear time-invariant (LTI) systems are bounded-input bounded-output (BIBO) stable if the region of convergence (ROC) in the s- and z-planes includes the -axis and unit circle , respectively. The s-plane applies to continuous-time systems, and the z-plane applies to discrete-time systems. But here’s the easy part: For causal systems, the property is poles in the left-half s-plane and poles inside the unit circle of the z-plane.

Convolving Rectangles

The convolution of two identical rectangular-shaped pulses or sequences results in a triangle. The triangle peak is at the integral of the signal or sum of the sequence squared. Figure 17-1 depicts the property graphically.

Figure 17-1: Convolving rectangles produce a triangle, both continuous and discrete.

The Convolution Theorem

The four (linear) convolution theorems are Fourier transform (FT), discrete-time Fourier transform (DTFT), Laplace transform (LT), and z-transform (ZT).Note: The discrete-time Fourier transform (DFT) doesn’t count here because circular convolution is a bit different from the others in this set.

These four theorems have the same powerful result: Convolution in the time domain can be reduced to multiplication in the respective domains. For and signal or impulse response, becomes , where the function arguments may be .

Frequency Response Magnitude

For the continuous- and discrete-time domains, the frequency response magnitude of an LTI system is related to pole-zero geometry. For continuous-time signals, you work in the s-domain; if the system is stable, you get the frequency response magnitude by evaluating along the -axis. For discrete-time signals, you work in the z-domain; if the system is stable, you get the frequency response magnitude by evaluating around the unit circle as . In both cases, frequency response magnitude nulling occurs if or passes near or over a zero, and magnitude response peaking occurs if or passes near a pole. The system can’t be stable if a pole is on or .

Convolution with Impulse Functions

When you convolve anything with or , you get that same anything back, but it’s shifted by or . Case in point:

Spectrum at DC

The direct current (DC), or average value, of the signal impacts the corresponding frequency spectrum at . In the discrete-time domain, the same result holds for sequence , except the periodicity of in the discrete-time domain makes the DC component at also appear at all multiples of .

Frequency Samples of N-point DFT

If you sample a continuous-time signal at rate samples per second to produce , then you can load N samples of into a discrete-time Fourier transform (DFT) — or a fast Fourier transform (FFT), for which N is a power of 2. The DFT points k correspond to these continuous-time frequency values:

Assuming that is a real signal, the useful DFT points run from 0 to N/2.

Integrator and Accumulator Unstable

The integrator system and accumulator system are unstable by themselves. Why? A pole at or a pole at isn’t good. But you can use both systems to create a stable system by placing them in a feedback configuration. Figure 17-2 shows stable systems built with the integrator and accumulator building blocks.

Figure 17-2: Making stable systems by using integrator (a) and accumulator (b) subsystems.

You can find the stable closed-loop system functions by doing the algebra:

The Spectrum of a Rectangular Pulse

The spectrum of a rectangular pulse signal or sequence (which is the frequency response if you view the signal as the impulse response of a LTI system) has periodic spectral nulls. The relationship for continuous and discrete signals is shown in Figure 17-3.

Figure 17-3: Spectrum magnitude of a rectangular pulse signal (a) and rectangular sequence (b).

Odd Half-Wave Symmetry and Fourier Series Harmonics

A periodic signal with odd half-wave symmetry, where is the period, has Fourier series representation consisting of only odd harmonics. If, for some constant A, , then the same property holds with the addition of a spectra line at (DC). The square wave and triangle waveforms are both odd half-wave symmetric to within a constant offset.