Should DSP Undergraduate Students Study z-Transform Regions of Convergence?
Not long ago I presented my 3-day DSP class to a group of engineers at Tektronix Inc. in Beaverton Oregon [1]. After I finished covering my material on IIR filters' z-plane pole locations and filter stability, one of the Tektronix engineers asked a question similar to:
"I noticed that you didn't discuss z-plane regions of convergence here. In my undergraduate DSP class we spent a lot of classroom and homework time on the ...
Implementing Impractical Digital Filters
This blog discusses a problematic situation that can arise when we try to implement certain digital filters. Occasionally in the literature of DSP we encounter impractical digital IIR filter block diagrams, and by impractical I mean block diagrams that cannot be implemented. This blog gives examples of impractical digital IIR filters and what can be done to make them practical.
Implementing an Impractical Filter: Example 1
Reference [1] presented the digital IIR bandpass filter...
Filter a Rectangular Pulse with no Ringing
To filter a rectangular pulse without any ringing, there is only one requirement on the filter coefficients: they must all be positive. However, if we want the leading and trailing edge of the pulse to be symmetrical, then the coefficients must be symmetrical. What we are describing is basically a window function.
Consider a rectangular pulse 32 samples long with fs = 1 kHz. Here is the Matlab code to generate the pulse:
N= 64; fs= 1000; % Hz sample...Dealing With Fixed Point Fractions
Fixed point fractional representation always gives me a headache because I screw it up the first time I try to implement an algorithm. The difference between integer operations and fractional operations is in the overflow. If the representation fits in the fixed point result, you can not tell the difference between fixed point integer and fixed point fractions. When integers overflow, they lose data off the most significant bits. When fractions overflow, they lose data off...
The DFT Output and Its Dimensions
The Discrete Fourier Transform, or DFT, converts a signal from discrete time to discrete frequency. It is commonly implemented as and used as the Fast Fourier Transform (FFT). This article will attempt to clarify the format of the DFT output and how it is produced.
Living in the real world, we deal with real signals. The data we typically sample does not have an imaginary component. For example, the voltage sampled by a receiver is a real value at a particular point in time. Let’s...
Amplitude modulation and the sampling theorem
I am working on the 11th and probably final chapter of Think DSP, which follows material my colleague Siddhartan Govindasamy developed for a class at Olin College. He introduces amplitude modulation as a clever way to sneak up on the Nyquist–Shannon sampling theorem.
Most of the code for the chapter is done: you can check it out in this IPython notebook. I haven't written the text yet, but I'll outline it here, and paste in the key figures.
Convolution...
Exponential Smoothing with a Wrinkle
Introduction
This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by providing a set of preprocessing filters to improve the resolution of the DFT. Because of the exponential nature of sinusoidal functions, they have special mathematical properties when exponential smoothing is applied to them. These properties are derived and explained in this blog article.
Basic Exponential Smoothing
Exponential smoothing is also known as...
Discrete-Time PLLs, Part 1: Basics
In this series of tutorials on discrete-time PLLs we will be focusing on Phase-Locked Loops that can be implemented in discrete-time signal proessors such as FPGAs, DSPs and of course, MATLAB.
Compressive Sensing - Recovery of Sparse Signals (Part 1)
The amount of data that is generated has been increasing at a substantial rate since the beginning of the digital revolution. The constraints on the sampling and reconstruction of digital signals are derived from the well-known Nyquist-Shannon sampling theorem...
Analytic Signal
In communication theory and modulation theory we always deal with two phases: In-phase (I) and Quadrature-phase (Q). The question that I will discuss in this blog is that why we use two phases and not more.
A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT
IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving an exact formula for the frequency of a complex tone in a DFT. It is basically a parallel treatment to the real case given in Exact Frequency Formula for a Pure Real Tone in a DFT. Since a real signal is the sum of two complex signals, the frequency formula for a single complex tone signal is a lot less complicated than for the real case.
Theoretical...Three Bin Exact Frequency Formulas for a Pure Complex Tone in a DFT
IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving exact formulas for the frequency of a complex tone in a DFT. This time it is three bin versions. Although the problem is similar to the two bin version in my previous blog article "A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT"[1], a slightly different approach is taken using linear algebra concepts. Because of an extra degree of freedom...
Generating Partially Correlated Random Variables
IntroductionIt is often useful to be able to generate two or more signals with specific cross-correlations. Or, more generally, we would like to specify an $\left(N \times N\right)$ covariance matrix, $\mathbf{R}_{xx}$, and generate $N$ signals which will produce this covariance matrix.
There are many applications in which this technique is useful. I discovered a version of this method while analysing radar systems, but the same approach can be used in a very wide range of...
Feedback Controllers - Making Hardware with Firmware. Part 7. Turbo-charged DSP Oscillators
This article will look at some DSP Sine-wave oscillators and will show how an FPGA with limited floating-point performance due to latency, can be persuaded to produce much higher sample-rate sine-waves of high quality.
Comparisons will be made between implementations on Intel Cyclone V and Cyclone 10 GX FPGAs. An Intel numerically controlled oscillator
Instantaneous Frequency Measurement
I would like to talk about the oft used method of measuring the carrier frequency in the world of Signal Collection and Characterization world. It is an elegant technique because of its simplicity. But, of course, with simplicity, there come drawbacks (sometimes...especially with this one!).
In the world of Radar detection and characterization, one of the key characteristics of interest is the carrier frequency of the signal. If the radar is pulsed, you will have a very wide bandwidth, a...
Deconvolution by least squares (Using the power of linear algebra in signal processing).
When we deal with our normal discrete signal processing operations, like FIR/IIR filtering, convolution, filter design, etc. we normally think of the signals as a constant stream of numbers that we put in a sequence
A Recipe for a Common Logarithm Table
IntroductionThis is an article that is a digression from trying to give a better understanding to the Discrete Fourier Transform (DFT).
A method for building a table of Base 10 Logarithms, also known as Common Logarithms, is featured using math that can be done with paper and pencil. The reader is assumed to have some familiarity with logarithm functions. This material has no dependency on the material in my previous blog articles.
If you were ever curious about how...
A Two Bin Solution
IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by showing an implementation of how the parameters of a real pure tone can be calculated from just two DFT bin values. The equations from previous articles are used in tandem to first calculate the frequency, and then calculate the amplitude and phase of the tone. The approach works best when the tone is between the two DFT bins in terms of frequency.
The Coding...Hidden Linear Algebra in DSP
Linear algebra (LA) is usually thought of as a blunt theoretical subject. However, LA is found hidden in many DSP algorithms used widely in practice.
An obvious clue in finding LA in DSP is the linearity assumption used in theoretical analysis of systems for modelling or design. A standard modelling example for this case would be linear time invariant (LTI) systems. LTI are usually used to model flat wireless communication channels. LTI systems are also used in the design of digital filter...
Bank-switched Farrow resampler
Bank-switched Farrow resampler Summary
A modification of the Farrow structure with reduced computational complexity.Compared to a conventional design, the impulse response is broken into a higher number of segments. Interpolation accuracy is achieved with a lower polynomial order, requiring fewer multiplications per output sample at the expense of a higher overall number of coefficients.
Example codeThis code snippet provides a Matlab / Octave implementation.And
Implementing Impractical Digital Filters
This blog discusses a problematic situation that can arise when we try to implement certain digital filters. Occasionally in the literature of DSP we encounter impractical digital IIR filter block diagrams, and by impractical I mean block diagrams that cannot be implemented. This blog gives examples of impractical digital IIR filters and what can be done to make them practical.
Implementing an Impractical Filter: Example 1
Reference [1] presented the digital IIR bandpass filter...
Bank-switched Farrow resampler
Bank-switched Farrow resampler SummaryA modification of the Farrow structure with reduced computational complexity.Compared to a conventional design, the impulse response is broken into a higher number of segments. Interpolation accuracy is achieved with a lower polynomial order, requiring fewer multiplications per output sample at the expense of a higher overall number of coefficients.
Example codeThis code snippet provides a Matlab / Octave implementation.And
Filtering Noise: The Basics (Part 1)
Introduction
Finding signals in the presence of noise is one of the fundamental quests of the discipline of signal processing. Noise is inherently random by nature, so a probability oriented approach is needed to develop a mathematical framework for filtering (i.e. removing/suppressing) noise. This framework or discipline, formally referred to as stochastic signal processing, is often taught in graduate level engineering programs and is covered from different perspectives in excellent...
Feedback Controllers - Making Hardware with Firmware. Part 2. Ideal Model Examples
Developing and Validating Simulation ModelsThis article will describe models for simulating the systems and controllers for the hardware emulation application described in Part 1 of the series.
- Part 1: Introduction
- Part 2: Ideal Model Examples
- Part 3: Sampled Data Aspects
- Part 4: Engineering of Evaluation Hardware
- Part 5:
Feedback Controllers - Making Hardware with Firmware. Part 4. Engineering of Evaluation Hardware
Following on from the previous abstract descriptions of an arbitrary circuit emulation application for low-latency feedback controllers, we now come to some aspects in the hardware engineering of an evaluation design from concept to first power-up. In due course a complete specification along with application examples will be maintained on the project website.- Part 1: Introduction
- Part 2:...
Hidden Linear Algebra in DSP
Linear algebra (LA) is usually thought of as a blunt theoretical subject. However, LA is found hidden in many DSP algorithms used widely in practice.
An obvious clue in finding LA in DSP is the linearity assumption used in theoretical analysis of systems for modelling or design. A standard modelling example for this case would be linear time invariant (LTI) systems. LTI are usually used to model flat wireless communication channels. LTI systems are also used in the design of digital filter...
Multilayer Perceptrons and Event Classification with data from CODEC using Scilab and Weka
For my first blog, I thought I would introduce the reader to Scilab [1] and Weka [2]. In order to illustrate how they work, I will put together a script in Scilab that will sample using the microphone and CODEC on your PC and save the waveform as a CSV file.
Deconvolution by least squares (Using the power of linear algebra in signal processing).
When we deal with our normal discrete signal processing operations, like FIR/IIR filtering, convolution, filter design, etc. we normally think of the signals as a constant stream of numbers that we put in a sequence
Algebra's Laws of Powers and Roots: Handle With Care
Recently, for entertainment, I tried to solve a puzzling algebra problem featured on YouTube [1]. In due course I learned that algebra’s $$(a^x)^y=a^{xy}\qquad\qquad\qquad\qquad\qquad(1)$$
Law of Powers identity is not always valid (not always true) if variable a is real and exponents x and y are complex-valued.
The fact that Eq. (1) can’t reliably be used with complex x and y exponents surprised me. And then I thought, “Humm, …what other of algebra’s identities may also...
Exponential Smoothing with a Wrinkle
IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by providing a set of preprocessing filters to improve the resolution of the DFT. Because of the exponential nature of sinusoidal functions, they have special mathematical properties when exponential smoothing is applied to them. These properties are derived and explained in this blog article.
Basic Exponential Smoothing
Exponential smoothing is also known as...
