An s-Plane to z-Plane Mapping Example
While surfing around the Internet recently I encountered the 's-plane to z-plane mapping' diagram shown in Figure 1. At first I thought the diagram was neat because it's a good example of the old English idiom: "A picture is worth a thousand words." However, as I continued to look at Figure 1 I began to detect what I believe are errors in the diagram.
Reader, please take a few moments to see if you detect any errors in Figure 1.
...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...
A Recipe for a Basic Trigonometry Table
IntroductionThis is an article that is give a better understanding to the Discrete Fourier Transform (DFT) by showing how to build a Sine and Cosine table from scratch. Along the way a recursive method is developed as a tone generator for a pure tone complex signal with an amplitude of one. Then a simpler multiplicative one. Each with drift correction factors. By setting the initial values to zero and one degrees and letting it run to build 45 values, the entire set of values needed...
Polar Coding Notes: Channel Combining and Channel Splitting
Channel Combining
Channel combining is a step that combines copies of a given B-DMC $W$ in a recursive manner to produce a vector channel $W_N : {\cal X}^N \to {\cal Y}^N$, where $N$ can be any power of two, $N=2^n, n\le0^{[1]}$.
The notation $u_1^N$ as shorthand for denoting a row vector $(u_1, \dots , u_N)$.
The vector channel $W_N$ is the virtual channel between the input sequence $u_1^N$ to a linear encoder and the output sequence $y^N_1$ of $N$...
DFT Graphical Interpretation: Centroids of Weighted Roots of Unity
IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by framing it in a graphical interpretation. The bin calculation formula is shown to be the equivalent of finding the center of mass, or centroid, of a set of points. Various examples are graphed to illustrate the well known properties of DFT bin values. This treatment will only consider real valued signals. Complex valued signals can be analyzed in a similar manner with...
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...
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...
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...Python scipy.signal IIR Filter Design Cont.
In the previous post the Python scipy.signal iirdesign function was disected. We reviewed the basics of filter specification and reviewed how to use the iirdesign function to design IIR filters. The previous post I only demonstrated low pass filter designs. The following are examples how to use the iirdesign function for highpass, bandpass, and stopband filters designs.
Highpass FilterThe following is a highpass filter design for the different filter...
Design of an anti-aliasing filter for a DAC
Overview
- Octaveforge / Matlab design script. Download: here
- weighted numerical optimization of Laplace-domain transfer function
- linear-phase design, optimizes vector error (magnitude and phase)
- design process calculates and corrects group delay internally
- includes sinc() response of the sample-and-hold stage in the ADC
- optionally includes multiplierless FIR filter
Digital-to-analog conversion connects digital...
Resolving 'Can't initialize target CPU' on TI C6000 DSPs - Part 1
Introduction
Today I am going to discuss some of the basics that can help prevent errors that frustrate some users. The information is directed toward TI C6000 family DSPs, but much of it also applies to other TI DSPs. In many cases they represent the user's first involvement with using Code Composer Studio [CCS] and a target board. It has been my experience that the primary cause of the "Can't initialize target CPU" error message and similar messages like "Error connecting to...
Compute Images/Aliases of CIC Interpolators/Decimators
Cascade-Integrator-Comb (CIC) filters are efficient fixed-point interpolators or decimators. For these filters, all coefficients are equal to 1, and there are no multipliers. They are typically used when a large change in sample rate is needed. This article provides two very simple Matlab functions that can be used to compute the spectral images of CIC interpolators and the aliases of CIC decimators.
1. CIC InterpolatorsFigure 1 shows three interpolate-by-M...
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...
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:...
Resolving 'Can't initialize target CPU' on TI C6000 DSPs - Part 1
Introduction
Today I am going to discuss some of the basics that can help prevent errors that frustrate some users. The information is directed toward TI C6000 family DSPs, but much of it also applies to other TI DSPs. In many cases they represent the user's first involvement with using Code Composer Studio [CCS] and a target board. It has been my experience that the primary cause of the "Can't initialize target CPU" error message and similar messages like "Error connecting to...
FIR sideways (interpolator polyphase decomposition)
An efficient implementation of a symmetric-FIR polyphase 1:3 interpolator that doesn't follow the usual tapped delay line-paradigm. The example exploits the impulse response symmetry and avoids four multiplications out of 10. keywords: symmetric polyphase FIR filter implementation ASIC Matlab / Octave implementation
IntroductionAn interpolating FIR filter can be implemented with a single tapped delay line, possibly going forwards and backwards for a symmetric impulse response. To...
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...
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...
Some Thoughts on Sampling
Some time ago, I came across an interesting problem. In the explanation of sampling process, a representation of impulse sampling shown in Figure 1 below is illustrated in almost every textbook on DSP and communications. The question is: how is it possible that during sampling, the frequency axis gets scaled by $1/T_s$ -- a very large number? For an ADC operating at 10 MHz for example, the amplitude of the desired spectrum and spectral replicas is $10^7$! I thought that there must be...
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
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
