Mark Wickert

Cheat Sheet / Updated 02-16-2022

Signals and systems is an aspect of electrical engineering that applies mathematical concepts to the creation of product design, such as cell phones and automobile cruise control systems. Absorbing the core concepts of signals and systems requires a firm grasp on their properties and classifications; a solid knowledge of algebra, trigonometry, complex arithmetic, calculus of one variable; and familiarity with linear constant coefficient (LCC) differential equations.

Step by Step / Updated 06-02-2016

You probably have some level of familiarity with consumer electronics, such as MP3 music players, smartphones, and tablet devices, and realize that these products rely on signals and systems. But you may take for granted the cruise control in your car. Here, the signals and systems framework in three familiar devices are shown at the block diagram level — a system diagram that identifies the significant components inside rectangular boxes, interconnected with arrows that show the direction of signal flow. The block diagram expresses the overall concept of a system without intimate implementation details.

Article / Updated 03-26-2016

Sampling theory links continuous and discrete-time signals and systems. For example, you can get a discrete-time signal from a continuous-time signal by taking samples every T seconds. This article points out some useful relationships associated with sampling theory. Key concepts include the low-pass sampling theorem, the frequency spectrum of a sampled continuous-time signal, reconstruction using an ideal lowpass filter, and the calculation of alias frequencies. The table of properties begins with a block diagram of a discrete-time processing subsystem that produces continuous-time output y(t) from continuous-time input x(t). This block diagram motivates the sampling theory properties in the remainder of the table. The process of converting continuous-time signal x(t) to discrete-time signal x[n] requires sampling, which is implemented by the analog-to-digital converter (ADC) block. The block with frequency response represents a linear time invariant system with input x[n] and output y[n]. The discrete-time signal y[n] is returned to the continuous-time domain via a digital-to-analog converter and a reconstruction filter.

Article / Updated 03-26-2016

Both signals and systems can be analyzed in the time-, frequency-, and s- and z-domains. Leaving the time-domain requires a transform and then an inverse transform to return to the time-domain. As you work to and from the time domain, referencing tables of both transform theorems and transform pairs can speed your progress and make the work easier. Use this table of common pairs for the continuous-time Fourier transform, discrete-time Fourier transform, the Laplace transform, and the z-transform as needed. Working in the frequency domain means you are working with Fourier transform and discrete-time Fourier transform — in the s-domain. Using Fourier transforms for continuous-time signals Here is a short table of theorems and pairs for the continuous-time Fourier transform (FT), in both frequency variable The forward and inverse transforms for these two notational schemes are defined as: . . . and here's the table: Applying Fourier transform to discrete-time signals For discrete-time signals and systems the discrete-time Fourier transform (DTFT) takes you to the frequency domain. A short table of theorems and pairs for the DTFT can make your work in this domain much more fun. The discrete-time frequency variable is The forward and inverse transforms are defined as: . . . and here's the table: Using the Laplace transform in the s-domain For continuous-time signals and systems, the one-sided Laplace transform (LT) helps to decipher signal and system behavior. It's also the best approach for solving linear constant coefficient differential equations with nonzero initial conditions. The one-sided LT is defined as: The inverse LT is typically found using partial fraction expansion along with LT theorems and pairs. Here's a short table of LT theorems and pairs. Letting the z-Transform help with signals and systems analysis For discrete-time signals and systems, the z-transform (ZT) is the counterpart to the Laplace transform. With the ZT you can characterize signals and systems as well as solve linear constant coefficient difference equations. The two-sided ZT is defined as: The inverse ZT is typically found using partial fraction expansion and the use of ZT theorems and pairs. Here is a short table of ZT theorems and pairs.

Article / Updated 03-26-2016

The study of signals and systems establishes a mathematical formalism for analyzing, modeling, and simulating electrical systems in the time, frequency, and s- or z-domains. Signals exist naturally and are also created by people. Some operate continuously (known as continuous-time signals); others are active at specific instants of time (and are called discrete-time signals). Signals pass through systems to be modified or enhanced in some way. Systems that operate on signals are also categorized as continuous- or discrete-time. Mathematics plays a central role in all facets of signals and systems. Specifically, complex arithmetic, trigonometry, and geometry are mainstays of this dynamic and (ahem) electrifying field of work and study. This article highlights the most applicable concepts from each of these areas of math for signals and systems work. Complex arithmetic for signals and systems Here are some of the most important complex arithmetic operations and formulas that relate to signals and systems. Trigonometry and Euler's formulas This table presents the key formulas of trigonometry that apply to signals and systems: Geometric series Among the most important geometry equations to know for signals and systems are these three:

Article / Updated 03-26-2016

Part of learning about signals and systems is that systems are identified according to certain properties they exhibit. Have a look at the core system classifications: Linearity: A linear combination of individually obtained outputs is equivalent to the output obtained by the system operating on the corresponding linear combination of inputs. Time-invariant: The system properties don't change with time. A present input produces the same response as it does in the future, less the time shift factor between the present and future. Memoryless: If the present system output depends only on the present input, the system is memoryless. Causal: The present system output depends at most on the present and past inputs. Future inputs can't be used to produce the present output. Stable: A system is bounded-input bound-output (BIBO) stable if all bounded inputs produce a bounded output. This table presents core linear time invariant (LTI) system properties for both continuous and discrete-time systems. Time-domain, frequency-domain, and s/z-domain properties are identified for the categories basic input/output, cascading, linear constant coefficient (LCC) differential and difference equations, and BIBO stability:

Article / Updated 03-26-2016

Periodic signals can be synthesized as a linear combination of harmonically related complex sinusoids. The theory of Fourier series provides the mathematical tools for this synthesis by starting with the analysis formula, which provides the Fourier coefficients Xn corresponding to periodic signal x(t) having period T0. Common periodic signals include the square wave, pulse train, and triangle wave. This table shows the Fourier series analysis and synthesis formulas and coefficient formulas for Xn in terms of waveform parameters for the provided waveform sketches:

Article / Updated 03-26-2016

Signals — both continuous-time signals and their discrete-time counterparts — are categorized according to certain properties, such as deterministic or random, periodic or aperiodic, power or energy, and even or odd. These traits aren't mutually exclusive; signals can hold multiple classifications. Here are some of the most important signal properties. But wait! There's more. Signals can also be categorized as exponential, sinusoidal, or a special sequence. The unit sample sequence and the unit step sequence are special signals of interest in discrete-time. All the continuous-time signal classifications have discrete-time counterparts, except singularity functions, which appear in continuous-time only. Defining special signals that serve as building blocks for more complex signals makes the creation of custom signal models to suit your needs more systematic and convenient.

Article / Updated 03-26-2016

Following are eleven signals and systems concepts that apply to the design of a signal processing system known as an audio graphic equalizer. When you listen to music on a portable music player or a computer, you can usually customize the sound— you can re-shape the frequency spectrum of the underlying music signal to suit your tastes using a set of ten tone controls. Establish core requirements of an equalizer As the signals and systems engineer, you need a better understanding of the frequency response of the listening halls targeted for the presets list. The shape of the frequency response that the equalizer must represent dictates the core requirements of the equalizer architecture. For example, the frequency response extremes are of special interest, as they will help establish the frequency response shape of the individual filters, the gain extremes needed per filter, and maybe the sampling rate. Select the system sampling rate The acoustics research reveals information about the frequency response shapes that the system must model. For a discrete-time implementation, you must choose the sampling rate to capture the relevant frequency bands of interest for audio signals. The sampling rate is most likely dictated by the audio playback sampling rate. For studio grade equipment, a sampling rate of 48, 96, or 128 kSPS is appropriate. For compact disc (CD) quality sound, the standard is 44.1 kHz. Excess sampling rate drives up the signal processing requirements as the sum total of samples per second and filtering requirements of multiplies and adds per filter come to bear on the implementation needs of the design. Choose a prototype filter You have to look at the basic filter type for the design in content of many filters acting roughly independent of one another. With each slider, you want the ability to raise and lower the volume of just one band of frequencies independent of the other bands, which you may want to hold fixed, with no level changes. You want the filter to be relatively simple because you’ll need many filter bands, but the filter needs to be easily tuned for the different band applications. The ideal filter is one that passes all adjacent frequency bands with unity gain yet can raise and lower the gain over a narrow band of frequencies. The filter of choice is known as the peaking filter. Decide how many filters are needed The frequency response plots you get as part of the acoustics research provides information about how many filter bands are needed. Practicality also comes into play here as does tradition. Tradition shows that ten octave band-spaced filters make a reasonably good audio equalizer design. In particular, octave band center frequencies spread from 31.25 Hz to 16,000 Hz, reasonably covering the 20 Hz to 20 kHz audio spectrum. Note the equalizer's system architecture A candidate architecture approach for the ten-band equalizer is to insert a cascade of ten digital peaking filters having system function Hi(z), i = 0, 1, . . . , 9, between the signal source and the digital-to-analog converter (DAC). Credit: Illustration by Mark Wickert, PhD Ideally, each filter has a frequency response magnitude that provides unity gain (0 dB because 20 log10[1] = 0) everywhere except in a narrow band of frequencies about the center frequency given under the sliders. The pass bands of the filters are contiguous so that the ten bands together approximately cover the audio spectrum from 20 Hz to 20 kHz. The idealized frequency response magnitude of the cascade is shown for a particular equalizer setting and sampling rate fs. Credit: Illustration by Mark Wickert, PhD Characterize the peaking filter A peaking filter provides gain or loss (attenuation) at a specific center frequency fc. A peaking filter has unity frequency response magnitude, or 0 dB gain, at frequencies far removed from the center frequency. At the center frequency fc, the frequency response magnitude in dB is GdB, which is continuously adjustable over a range of, say, +/–12 dB. Before committing to a final design, you need additional characterization. At the same time, the filter complexity is revealed, because studying the frequency response means you’ll likely need to work from the system function. From the system function, you can also arrive at the difference equation representation, which is closely related to a filter implementation algorithm. Choose the peaking filter Q-value The parameter Q is inversely proportional to the filter bandwidth. For a fixed Q, 3.5 ripples occur between the octave bands. The ability to implement peaks and valleys in the overall frequency response traces back to the acoustics research and also to the need to cover ten octaves from 31.25 Hz to 16 kHz. If Q is too large, the ripples in the cascade frequency response mean that some frequencies can’t be gain controlled at all, but if Q is too small, the individual pass bands bleed together, making it more difficult to represent frequency response detail. A Q of 3.5 is chosen for the remainder of this design analysis. Consider gain adjustment range The acoustics research and equalizer preset needs dictate the range of gain value needed for each slider. As a practical matter, too wide a range of gain values make the processing algorithms more complex because of dynamic range considerations. A reasonable starting point is +/–12 dB. Note that CD audio is recorded with 16-bits of dynamic range, which corresponds to about 96 dB of total signal dynamic range. Allowing an equalizer to raise and lower the gain of individual frequency bands by +/–12 dB adds an additional 24-dB dynamic range onto the system. You need more bits of precision in the output signal stream to make this viable in the system DAC. Work the algorithm math When implementing discrete-time systems, you have the choice of using floating-point arithmetic or fixed-point arithmetic for the filter difference equations. Fixed-point math is generally more efficient, depending on the processor architecture. If the processor supports floating-point operations, floating-point is the way to go. But in small battery power devices, floating-point may not be available. You need to study the peaking filters to see whether all ten octaves can be readily built by using fixed-point math without incurring any performance penalties. When using fixed-point math, the bit width is generally 16-bit signed numbers, while floating-point is generally 32 bits. Exercise the equalizer with test bed considerations You need to conduct a so-called bit true arithmetic test bed to fully exercise the equalizer with real test signals from standard music sources. The idea with the test bed is to be able to evaluate the complete system performance with real signals and real equalizer presets that the customer has asked for. With the test bed, you should be able to exercise all critical requirements of the design. Test performance The real deal with the ten-band equalizer is that you can graphically visualize the spectral shaping you provide to the signal passing through the equalizer by just looking at the positions of slider gain controls. Credit: Illustration by Mark Wicker, PhD

Article / Updated 03-26-2016

Options a and b are the fixed FIR and IIR notch filters, respectively. The simplicity of these filters is a major draw. But how well do they work? Characterizing the filters in the frequency domain is a good starting point for this assessment. A sinusoidal signal of the form won’t pass through these filters in steady state. The following figure provides magnitude response in dB versus frequency plots of the FIR and IIR notch filters, when fi = 1,000 Hz and fs = 8,000 Hz. For the IIR, notch r is stepped over 0.8, 0.9, and 0.95. In Figure c, a cascade of two IIR notch filters having r = 0.95 is made for fi = 1,000 and 600 Hz. Here are the IPython command line inputs for creating the IIR cascade: Credit: Illustration by Mark Wickert, PhD In [<b>659</b>]: bIIR95, aIIR95 = ssd.fir_iir_notch(1000,8000,0.95) In [<b>664</b>]: bIIR95_600,aIIR95_600 = ssd.fir_iir_notch(600,8000,0.95) In [<b>665</b>]: bIIR_cas,aIIR_cas = ssd.cascade_filters(bIIR95,aIIR95, bIIR95_600,aIIR95_600) In [<b>666</b>]: f,HIIR_cas = signal.freqz(bIIR_cas,aIIR_cas,1024) SNOI tones are blocked if is set properly. What spectral components from the SOI are also removed? Unfortunately, some SOI information is lost, but you want to minimize the loss within reason. The FIR notch, although simple to implement, removes too much information from the SOI. So drop this filter from further consideration. Going with the IIR notch is worth the extra effort to get the precision removal of essentially just the SNOI. You have to accept that if the SOI has frequency components at the SNOI frequencies, they, too, will be removed by the FIR and IIR filters. The adaptive filter comes with many parameters to play with. For this study, the filter length is M = 64 and μ = 0.005.