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

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.

## 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 *x*_{1} and *x*_{2} signal or impulse response, y = *x*_{1 * }*x*_{2}_{ }becomes

## 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 *j**ω*-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

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:

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

## Convolution with impulse functions

When you convolve *anything* with

you get that same anything back, but it’s shifted by *t*_{0} or *n*_{0}. Case in point:

## 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

in the discrete-time domain makes the DC component at

## Frequency samples of N-point DFT

If you sample a continuous-time signal *x*(*t*) at rate *f** _{s}* samples per second to produce

*x*[

*n*] =

*x*(

*n*/

*f*

*), then you can load*

_{s}*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:

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 *H** _{i}*(

*s*) = 1/

*s*and accumulator system

*H*

_{acc}(

*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.

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 here.

## Odd half-wave symmetry and Fourier series harmonics

A periodic signal with odd half-wave symmetry,

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.