## A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT

●1 commentIntroduction This 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...

## DFT Bin Value Formulas for Pure Complex Tones

Introduction This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving an analytical formula for the DFT of pure complex tones and an alternative variation. It is basically a parallel...

## Multi-Decimation Stage Filtering: Design and Optimization

●1 commentDuring my research on digital FIR decimation filters I have been developing various Matlab scripts and functions. In which I have decided later on to consolidate it in a form of a toolbox. I have developed this toolbox to assist and...

## Canonic Signed Digit (CSD) Representation of Integers

●1 commentIn my last post I presented Matlab code to synthesize multiplierless FIR filters using Canonic Signed Digit (CSD) coefficients. I included a function dec2csd1.m (repeated here in Appendix A) to convert decimal integers to binary CSD...

## Frequency Translation by Way of Lowpass FIR Filtering

●1 commentSome weeks ago a question appeared on the dsp.related Forum regarding the notion of translating a signal down in frequency and lowpass filtering in a single operation [1]. It is possible to implement such a process by embedding a discrete cosine...

## Minimum Shift Keying (MSK) - A Tutorial

Minimum Shift Keying (MSK) is one of the most spectrally efficient modulation schemes available. Due to its constant envelope, it is resilient to non-linear distortion and was therefore chosen as the modulation technique for the GSM cell phone...

## Round Round Get Around: Why Fixed-Point Right-Shifts Are Just Fine

●1 commentToday’s topic is rounding in embedded systems, or more specifically, why you don’t need to worry about it in many cases.One of the issues faced in computer arithmetic is that exact arithmetic requires an ever-increasing bit length to...

## Some Thoughts on Sampling

●1 commentSome 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...

## Matlab Code to Synthesize Multiplierless FIR Filters

This article presents Matlab code to synthesize multiplierless Finite Impulse Response (FIR) lowpass filters. A filter coefficient can be represented as a sum of powers of 2. For example, if a coefficient = decimal 5 multiplies input x,...

## Wavelets II - Vanishing Moments and Spectral Factorization

●1 commentIn the previous blog post I described the workings of the Fast Wavelet Transform (FWT) and how wavelets and filters are related. As promised, in this article we will see how to construct useful filters. Concretely, we will find a way to calculate...

## 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...

## Discrete Wavelet Transform Filter Bank Implementation (part 1)

●1 commentUPDATE: Added graphs and code to explain the frequency division of the branches The focus of this article is to briefly explain an implementation of this transform and several filter bank forms. Theoretical information about DWT can be found...

## Free DSP Books on the Internet

●2 commentsWhile 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...

## Delay estimation by FFT

●3 commentsGiven x=sig(t) and y=ref(t), returns [c, ref(t+delta), delta)] = fitSignal(y, x);:Estimates and corrects delay and scaling factor between two signals Code snippet This article relates to the Matlab / Octave code snippet: Delay estimation with...

## 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...

## Python scipy.signal IIR Filtering: An Example

Introduction In the last posts I reviewed how to use the Python scipy.signal package to design digital infinite impulse response (IIR) filters, specifically, using the iirdesign function (IIR design I and IIR design...

## Stereophonic Amplitude-Panning: A Derivation of the "Tangent Law"

This article presents a derivation of the "Tangent Law"

## A Brief Introduction To Romberg Integration

This article briefly describes a remarkable integration algorithm, called "Romberg integration." The algorithm is used in the field of numerical analysis but it's not so well-known in the world of DSP.

## Sinusoidal Frequency Estimation Based on Time-Domain Samples

●6 commentsThe topic of estimating a noise-free real or complex sinusoid's frequency, based on fast Fourier transform (FFT) samples, has been presented in recent blogs here on dsprelated.com. For completeness, it's worth knowing that simple frequency estimation algorithms exist that do not require FFTs to be performed . Below I present three frequency estimation algorithms that use time-domain samples, and illustrate a very important principle regarding so called "exact" mathematically-derived DSP algorithms.