## Decimator Image Response

This article presents a way to compute and plot the image response of a decimator. I'm defining the image response as the unwanted spectrum of the impulse response after downsampling, relative to the desired passband response.

## 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.

## Digital Envelope Detection: The Good, the Bad, and the Ugly

●12 commentsRecently I've been thinking about the process of envelope detection. Tutorial information on this topic is readily available but that information is spread out over a number of DSP textbooks and many Internet web sites. The purpose of this blog is to summarize various digital envelope detection methods in one place. Here I focus of envelope detection as it is applied to an amplitude-fluctuating sinusoidal signal where the positive-amplitude fluctuations (the sinusoid's envelope) contain some sort of information. Let's begin by looking at the simplest envelope detection method.

## Python For Audio Signal Processing

●6 commentsThis paper discusses the use of Python for developing audio signal processing applications. Overviews of Python language, NumPy, SciPy and Matplotlib are given, which together form a powerful platform for scientific computing. We then show how SciPy was used to create two audio programming libraries, and describe ways that Python can be integrated with the SndObj library and Pure Data, two existing environments for music composition and signal processing.

## Lecture Notes on Elliptic Filter Design

●1 commentElliptic filters, also known as Cauer or Zolotarev filters, achieve the smallest filter order for the same specifications, or, the narrowest transition width for the same filter order, as compared to other filter types. On the negative side, they have the most nonlinear phase response over their passband. In these notes, we are primarily concerned with elliptic filters. But we will also discuss briefly the design of Butterworth, Chebyshev-1, and Chebyshev-2 filters and present a unified method of designing all cases. We also discuss the design of digital IIR filters using the bilinear transformation method.

## Optimizing the Half-band Filters in Multistage Decimation and Interpolation

●2 commentsThis article discusses a not so well-known rule regarding the filtering in multistage decimation and interpolation by an integer power of two.

## The DFT Magnitude of a Real-valued Cosine Sequence

●5 commentsThis article may seem a bit trivial to some readers here but, then again, it might be of some value to DSP beginners. It presents a mathematical proof of what is the magnitude of an N-point discrete Fourier transform (DFT) when the DFT's input is a real-valued sinusoidal sequence.

## Sum of Two Equal-Frequency Sinusoids

●4 commentsThe sum of two equal-frequency real sinusoids is itself a single real sinusoid. However, the exact equations for all the various forms of that single equivalent sinusoid are difficult to find in the signal processing literature. Here we provide those equations.

## Using the DFT as a Filter: Correcting a Misconception

●1 commentI have read, in some of the literature of DSP, that when the discrete Fourier transform (DFT) is used as a filter the process of performing a DFT causes an input signal's spectrum to be frequency translated down to zero Hz (DC). I can understand why someone might say that, but I challenge that statement as being incorrect. Here are my thoughts.

## Negative Group Delay

●2 commentsDispersive linear systems with negative group delay have caused much confusion in the past. Some claim that they violate causality, others that they are the cause of superluminal tunneling. Can we really receive messages before they are sent? This article aims at pouring oil in the fire and causing yet more confusion :-).

## An Efficient Linear Interpolation Scheme

●4 commentsThis article presents a computationally-efficient linear interpolation trick that requires at most one multiply per output sample.

## 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.

## Digital Signal Processor Fundamentals and System Design

●9 commentsDigital Signal Processors (DSPs) have been used in accelerator systems for more than fifteen years and have largely contributed to the evolution towards digital technology of many accelerator systems, such as machine protection, diagnostics and control of beams, power supply and motors. This paper aims at familiarising the reader with DSP fundamentals, namely DSP characteristics and processing development. Several DSP examples are given, in particular on Texas Instruments DSPs, as they are used in the DSP laboratory companion of the lectures this paper is based upon. The typical system design flow is described; common difficulties, problems and choices faced by DSP developers are outlined; and hints are given on the best solution.

## The Risk In Using Frequency Domain Curves To Evaluate Digital Integrator Performance

This article shows the danger in evaluating the performance of a digital integration network based solely on its frequency response curve. If you plan on implementing a digital integrator in your signal processing work I recommend you continue reading this article.

## Digital PLL's - Part 2

●1 commentIn Part 1, we found the time response of a 2nd order PLL with a proportional + integral (lead-lag) loop filter. Now let's look at this PLL in the Z-domain.

## The World's Most Interesting FIR Filter Equation: Why FIR Filters Can Be Linear Phase

●9 commentsThis article discusses a little-known filter characteristic that enables real- and complex-coefficient tapped-delay line FIR filters to exhibit linear phase behavior. That is, this article answers the question: What is the constraint on real- and complex-valued FIR filters that guarantee linear phase behavior in the frequency domain?

## The Swiss Army Knife of Digital Networks

●4 commentsThis article describes a general discrete-signal network that appears, in various forms, inside so many DSP applications.

## C++ Tutorial

●12 commentsThis tutorial is for those people who want to learn programming in C++ and do not necessarily have any previous knowledge of other programming languages. Of course any knowledge of other programming languages or any general computer skill can be useful to better understand this tutorial, although it is not essential. It is also suitable for those who need a little update on the new features the language has acquired from the latest standards. If you are familiar with the C language, you can take the first 3 parts of this tutorial as a review of concepts, since they mainly explain the C part of C++. There are slight differences in the C++ syntax for some C features, so I recommend you its reading anyway. The 4th part describes object-oriented programming. The 5th part mostly describes the new features introduced by ANSI-C++ standard.

## Sum of Two Equal-Frequency Sinusoids

●4 commentsThe sum of two equal-frequency real sinusoids is itself a single real sinusoid. However, the exact equations for all the various forms of that single equivalent sinusoid are difficult to find in the signal processing literature. Here we provide those equations.