## Sum of Two Equal-Frequency Sinusoids

The 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

I 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

Dispersive 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 :-).

## Implementing Simultaneous Digital Differentiation, Hilbert Transformation, and Half-Band Filtering

Recently I've been thinking about digital differentiator and Hilbert transformer implementations and I've developed a processing scheme that may be of interest to the readers here on dsprelated.com.

## A New Contender in the Digital Differentiator Race

This blog proposes a novel differentiator worth your consideration. Although simple, the differentiator provides a fairly wide 'frequency range of linear operation' and can be implemented, if need be, without performing numerical multiplications.

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

This 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?

## Correcting an Important Goertzel Filter Misconception

Correcting an Important Goertzel Filter Misconception

## Complex Down-Conversion Amplitude Loss

This article illustrates the signal amplitude loss inherent in a traditional complex down-conversion system. (In the literature of signal processing, complex down-conversion is also called "quadrature demodulation.")

## Specifying the Maximum Amplifier Noise When Driving an ADC

I recently learned an interesting rule of thumb regarding the use of an amplifier to drive the input of an analog to digital converter (ADC). The rule of thumb describes how to specify the maximum allowable noise power of the amplifier.

## Digital Filtering in the Frequency Domain

Time domain digital filtering, whether implemented using FIR or IIR techniques, has been very well documented in literature and been thoroughly used in many base band processor designs. However, with the advent of software defined radios as well as CPU support in more recent baseband processors, it has become possible and often desirable to filter signals in software rather than digital hardware. Whereas, time domain digital filtering can certainly be implemented in software as well, it becomes highly inefficient as the number of filter taps grows. Frequency domain filtering, using FFT and IFFT operations, is significantly more efficient and surprisingly easy to understand. This document introduces the reader to frequency domain filtering both in theory and in practice via a MatLab script.

## Complex Down-Conversion Amplitude Loss

This article illustrates the signal amplitude loss inherent in a traditional complex down-conversion system. (In the literature of signal processing, complex down-conversion is also called "quadrature demodulation.")

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

This 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.

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

I 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.

## Cascaded Integrator-Comb (CIC) Filter Introduction

In the classic paper, "An Economical Class of Digital Filters for Decimation and Interpolation", Hogenauer introduced an important class of digital filters called "Cascaded Integrator-Comb", or "CIC" for short (also sometimes called "Hogenauer filters"). Here, Matthew Donadio provides a more gentle introduction to the subject of CIC filters, geared specifically to the needs of practicing DSP designers.

## Reduced-Delay IIR Filters

This document describes a straightforward method to significantly reduce the number of necessary multiplies per input sample of traditional IIR lowpass and highpass digital filters.

## An Experimental Multichannel Pulse Code Modulation System of Toll Quality + Electron Beam Deflection Tube For Pulse Code Modulation

See this blog post for context. Pulse Code Modulation offers attractive possibilities for multiplex telephony via such media as the microwave radio relay. The various problems involved in its use have been explored in terms of a 96-channel system designed to meet the transmission requirements commonly imposed upon commercial toll circuits. Twenty-four of the 96 channels have been fully equipped in an experimental model of the system. Coding and decoding devices are described, along with other circuit details. The coder is based upon a new electron beam tube, and is characterized by speed and simplicity as well as accuracy of coding. These qualities are matched in the decoder, which employs pulse excitation of a simple reactive network.

## Sum of Two Equal-Frequency Sinusoids

The 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.

## Implementing Simultaneous Digital Differentiation, Hilbert Transformation, and Half-Band Filtering

Recently I've been thinking about digital differentiator and Hilbert transformer implementations and I've developed a processing scheme that may be of interest to the readers here on dsprelated.com.

## The DFT of Finite-Length Time-Reversed Sequences

Recently I've been reading papers on underwater acoustic communications systems and this caused me to investigate the frequency-domain effects of time-reversal of time-domain sequences. I created this article because there is so little coverage of this topic in the literature of DSP.