A Fast Real-Time Trapezoidal Rule Integrator
This article presents a computationally-efficient network for computing real?time discrete integration using the Trapezoidal Rule.
Using Mason's Rule to Analyze DSP Networks
There have been times when I wanted to determine the z-domain transfer function of some discrete network, but my algebra skills failed me. Some time ago I learned Mason's Rule, which helped me solve my problems. If you're willing to learn the...
The Discrete Fourier Transform as a Frequency Response
The discrete frequency response H(k) of a Finite Impulse Response (FIR) filter is the Discrete Fourier Transform (DFT) of its impulse response h(n) [1]. So, if we can find H(k) by whatever method, it should be identical to the DFT of...
Multiplierless Exponential Averaging
This blog discusses an interesting approach to exponential averaging. To begin my story, a traditional exponential averager (also called a "leaky integrator"), shown in Figure 1(a), is commonly used to reduce noise fluctuations that contaminate...
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...
Add the Hilbert Transformer to Your DSP Toolkit, Part 1
In some previous articles, I made use of the Hilbert transformer, but did not explain its theory in any detail. In this article, I’ll dig a little deeper into how the Hilbert Transformer works. Understanding the Hilbert Transformer...
A Differentiator With a Difference
Some time ago I was studying various digital differentiating networks, i.e., networks that approximate the process of taking the derivative of a discrete time-domain sequence. By "studying" I mean that I was experimenting with various...
Find Aliased ADC or DAC Harmonics (with animation)
When a sinewave is applied to a data converter (ADC or DAC), device nonlinearities produce harmonics. If a harmonic frequency is greater than the Nyquist frequency, the harmonic appears as an alias. In this case, it is not at once...
Polyphase Filters and Filterbanks
ALONG CAME POLY Polyphase filtering is a computationally efficient structure for applying resampling and filtering to a signal. Most digital filters can be applied in a polyphase format, and it is also possible to create efficient resampling...
An Efficient Full-Band Sliding DFT Spectrum Analyzer
In this blog I present two computationally efficient full-band discrete Fourier transform (DFT) networks that compute the 0th bin and all the positive-frequency bin outputs for an N-point DFT in real-time on a sample-by-sample basis. An Even-N...
Add a Power Marker to a Power Spectral Density (PSD) Plot
Perhaps we should call most Power Spectral Density (PSD) calculations relative PSD, because usually we don’t have to worry about absolute power levels. However, for cases (e.g., measurements or simulations) where we are concerned with...
A Simpler Goertzel Algorithm
In this blog I propose a Goertzel algorithm that is simpler than the version of the Goertzel algorithm that is traditionally presented DSP textbooks. Below I very briefly describe the DSP textbook version of the Goertzel algorithm followed by a...
60-Hz Noise and Baseline Drift Reduction in ECG Signal Processing
Electrocardiogram (ECG) signals are obtained by monitoring the electrical activity of the human heart for medical diagnostic purposes [1]. This blog describes a very efficient digital filter used to reduce both 60 Hz AC powerline noise and...
Find Aliased ADC or DAC Harmonics (with animation)
When a sinewave is applied to a data converter (ADC or DAC), device nonlinearities produce harmonics. If a harmonic frequency is greater than the Nyquist frequency, the harmonic appears as an alias. In this case, it is not at once...
Adaptive Beamforming is like Squeezing a Water Balloon
Adaptive beamforming was first developed in the 1960s for radar and sonar applications. The main idea is that signals can be captured using multiple sensors and the sensor outputs can be combined to enhance the signals propagating from...
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...
Exploring Human Hearing Range
Human Hearing Range In this post, I'll look at an interesting aspect of Audacity – using it to explore the threshold of human hearing. In my book Digital Signal Processing: A Gentle Introduction with Audio Examples, I go into this topic...
The Zeroing Sine Family of Window Functions
Introduction This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by introducing a class of well behaved window functions that the author believes to be previously unrecognized. The definition...
Design Square-Root Nyquist Filters
In his book on multirate signal processing, harris presents a nifty technique for designing square-root Nyquist FIR filters with good stopband attenuation [1]. In this post, I describe the method and provide a Matlab function for designing the filters. You can find a Matlab function by harris for designing the filters at [2].
How to Find a Fast Floating-Point atan2 Approximation
Context Over a short period of time, I came across nearly identical approximations of the two parameter arctangent function, atan2, developed by different companies, in different countries, and even in different decades. Fascinated...
Free DSP Books on the Internet
While surfing the "net" I have occasionally encountered signal processing books whose chapters could be downloaded to my computer. I started keeping a list of those books and, over the years, that list has grown to over forty books. Perhaps the...
A Fixed-Point Introduction by Example
Introduction The finite-word representation of fractional numbers is known as fixed-point. Fixed-point is an interpretation of a 2's compliment number usually signed but not limited to sign representation. It...
Decimators Using Cascaded Multiplierless Half-band Filters
In my last post, I provided coefficients for several multiplierless half-band FIR filters. In the comment section, Rick Lyons mentioned that such filters would be useful in a multi-stage decimator. For such an application, any subsequent multipliers save on resources, since they operate at a fraction of the maximum sample frequency. We’ll examine the frequency response and aliasing of a multiplierless decimate-by-8 cascade in this article, and we’ll also discuss an interpolator cascade using the same half-band filters.
Add the Hilbert Transformer to Your DSP Toolkit, Part 1
In some previous articles, I made use of the Hilbert transformer, but did not explain its theory in any detail. In this article, I’ll dig a little deeper into how the Hilbert Transformer works. Understanding the Hilbert Transformer...
Add the Hilbert Transformer to Your DSP Toolkit, Part 2
In this part, I’ll show how to design a Hilbert Transformer using the coefficients of a half-band filter as a starting point, which turns out to be remarkably simple. I’ll also show how a half-band filter can be synthesized using the...
The Discrete Fourier Transform and the Need for Window Functions
The Discrete Fourier Transform (DFT) is used to find the frequency spectrum of a discrete-time signal. A computationally efficient version called the Fast Fourier Transform (FFT) is normally used to calculate the DFT. But, as many...
A Simpler Goertzel Algorithm
In this blog I propose a Goertzel algorithm that is simpler than the version of the Goertzel algorithm that is traditionally presented DSP textbooks. Below I very briefly describe the DSP textbook version of the Goertzel algorithm followed by a...
Phase or Frequency Shifter Using a Hilbert Transformer
In this article, we'll describe how to use a Hilbert transformer to make a phase shifter or frequency shifter. In either case, the input is a real signal and the output is a real signal. We'll use some simple Matlab code to simulate these systems. After that, we'll go into a little more detail on Hilbert transformer theory and design.
Simplest Calculation of Half-band Filter Coefficients
Half-band filters are lowpass FIR filters with cut-off frequency of one-quarter of sampling frequency fs and odd symmetry about fs/4 [1]*. And it so happens that almost half of the coefficients are zero. The passband and stopband bandwiths are equal, making these filters useful for decimation-by-2 and interpolation-by-2. Since the zero coefficients make them computationally efficient, these filters are ubiquitous in DSP systems. Here we will compute half-band coefficients using the window method. While the window method typically does not yield the fewest taps for a given performance, it is useful for learning about half-band filters. Efficient equiripple half-band filters can be designed using the Matlab function firhalfband [2].
