Signals and Systems of 3 Familiar Devices
The Two-Sided z-Transform
AM Broadcast Frequency Spectrum

10 Signals and Systems Properties You Never Want to Forget

A big wide world of properties is associated with signals and systems — plenty in the math alone! Here are ten unforgettable 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

image0.jpg

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.

image1.jpg

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 x1 and x2 signal or impulse response, y = x1 * x2 becomes

image2.png

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 |H(s)| along the jω-axis.

For discrete-time signals, you work in the z-domain; if the system is stable, you get the frequency response magnitude by evaluating |H(z)| around the unit circle as

image3.jpg

In both cases, frequency response magnitude nulling occurs if either of the following values passes near or over a zero, and magnitude response peaking occurs if either of the following values passes near a pole:

image4.jpg

The system can’t be stable if a pole is on either value.

Convolution with impulse functions

When you convolve anything with

image5.jpg

you get that same anything back, but it’s shifted by t0 or n0. Case in point:

image6.jpg

Spectrum at DC

The direct current (DC), or average value, of the signal x(t) impacts the corresponding frequency spectrum X(f) at f = 0. In the discrete-time domain, the same result holds for sequence x[n], except the periodicity of

image7.jpg

in the discrete-time domain makes the DC component at

image8.jpg

Frequency samples of N-point DFT

If you sample a continuous-time signal x(t) at rate fs samples per second to produce x[n] = x(n/fs), then you can load N samples of x[n] 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:

image9.jpg

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

Integrator and accumulator unstable

The integrator system Hi(s) = 1/s and accumulator system Hacc(z) = 1/(1 – z–1) are unstable by themselves. Why? A pole at s = 0 or a pole at z = 1 isn’t good. But you can use both systems to create a stable system by placing them in a feedback configuration. This figure shows stable systems built with the integrator and accumulator building blocks.

image10.jpg

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

image11.jpg

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 here.

image12.jpg

Odd half-wave symmetry and Fourier series harmonics

A periodic signal with odd half-wave symmetry,

image13.jpg

is the period, has Fourier series representation consisting of only odd harmonics. If, for some constant A, y(t) = A + x(t), then the same property holds with the addition of a spectra line at f = 0 (DC). The square wave and triangle waveforms are both odd half-wave symmetric to within a constant offset.

  • Add a Comment
  • Print
  • Share
blog comments powered by Disqus
Create a System Block Diagram for the CD/DVD Case Study
Real-World Signals and Systems Case: Solving the DAC ZOH Droop Problem in the z-Domain
Real-World Signals and Systems Case: Analog Filter Design with a Twist
Testing Product Concepts with Behavioral Level Modeling
Implement a Real-World System: Karaoke Machine
Advertisement

Inside Dummies.com