## 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 :-).

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

●2 commentsRecently 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

●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?

## Correcting an Important Goertzel Filter Misconception

Correcting an Important Goertzel Filter Misconception

## Reducing IIR Filter Computational Workload

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

## 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.

## Decimation and Interpolation with Polyphase Filters

This article deals with decimation and interpolation in classical and polyphase realizations. The topic is accompanied by many simulations with MATLAB and Simulink, so that the facts can be easily fixed. The simulations can be creatively expanded with your own ideas.

## A Review of Physical and Perceptual Feature Extraction Techniques for Speech, Music and Environmental Sounds

●3 commentsEndowing machines with sensing capabilities similar to those of humans is a prevalent quest in engineering and computer science. In the pursuit of making computers sense their surroundings, a huge effort has been conducted to allow machines and computers to acquire, process, analyze and understand their environment in a human-like way. Focusing on the sense of hearing, the ability of computers to sense their acoustic environment as humans do goes by the name of machine hearing. To achieve this ambitious aim, the representation of the audio signal is of paramount importance. In this paper, we present an up-to-date review of the most relevant audio feature extraction techniques developed to analyze the most usual audio signals: speech, music and environmental sounds. Besides revisiting classic approaches for completeness, we include the latest advances in the field based on new domains of analysis together with novel bio-inspired proposals. These approaches are described following a taxonomy that organizes them according to their physical or perceptual basis, being subsequently divided depending on the domain of computation (time, frequency, wavelet, image-based, cepstral, or other domains). The description of the approaches is accompanied with recent examples of their application to machine hearing related problems.

## Peak-to-Average Power Ratio and CCDF

●1 commentPeak to Average Power Ratio (PAPR) is often used to characterize digitally modulated signals. One example application is setting the level of the signal in a digital modulator. Knowing PAPR allows setting the average power to a level that is just low enough to minimize clipping.

## Complex Digital Signal Processing in Telecommunications

●3 commentsDigital Signal Processing (DSP) is a vital tool for scientists and engineers, as it is of fundamental importance in many areas of engineering practice and scientific research. The "alphabet" of DSP is mathematics and although most practical DSP problems can be solved by using real number mathematics, there are many others which can only be satisfactorily resolved or adequately described by means of complex numbers. If real number mathematics is the language of real DSP, then complex number mathematics is the language of complex DSP. In the same way that real numbers are a part of complex numbers in mathematics, real DSP can be regarded as a part of complex DSP (Smith, 1999). Complex mathematics manipulates complex numbers - the representation of two variables as a single number - and it may appear that complex DSP has no obvious connection with our everyday experience, especially since many DSP problems are explained mainly by means of real number mathematics. Nonetheless, some DSP techniques are based on complex mathematics, such as Fast Fourier Transform (FFT), z-transform, representation of periodical signals and linear systems, etc. However, the imaginary part of complex transformations is usually ignored or regarded as zero due to the inability to provide a readily comprehensible physical explanation. One well-known practical approach to the representation of an engineering problem by means of complex numbers can be referred to as the assembling approach: the real and imaginary parts of a complex number are real variables and individually can represent two real physical parameters. Complex math techniques are used to process this complex entity once it is assembled. The real and imaginary parts of the resulting complex variable preserve the same real physical parameters. This approach is not universally-applicable and can only be used with problems and applications which conform to the requirements of complex math techniques. Making a complex number entirely mathematically equivalent to a substantial physical problem is the real essence of complex DSP. Like complex Fourier transforms, complex DSP transforms show the fundamental nature of complex DSP and such complex techniques often increase the power of basic DSP methods. The development and application of complex DSP are only just beginning to increase and for this reason some researchers have named it theoretical DSP. It is evident that complex DSP is more complicated than real DSP. Complex DSP transforms are highly theoretical and mathematical; to use them efficiently and professionally requires a large amount of mathematics study and practical experience. Complex math makes the mathematical expressions used in DSP more compact and solves the problems which real math cannot deal with. Complex DSP techniques can complement our understanding of how physical systems perform but to achieve this, we are faced with the necessity of dealing with extensive sophisticated mathematics. For DSP professionals there comes a point at which they have no real choice since the study of complex number mathematics is the foundation of DSP.

## A New Approach to Linear Filtering and Prediction Problems

●3 commentsIn 1960, R.E. Kalman published his famous paper describing a recursive solution to the discrete-data linear filtering problem. Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation.

## 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?

## Generating Complex Baseband and Analytic Bandpass Signals

●3 commentsThere are so many different time- and frequency-domain methods for generating complex baseband and analytic bandpass signals that I had trouble keeping those techniques straight in my mind. Thus, for my own benefit, I created a kind of reference table showing those methods. I present that table for your viewing pleasure in this document.

## How Discrete Signal Interpolation Improves D/A Conversion

●2 commentsEarlier this year, for the Linear Audio magazine, published in the Netherlands whose subscribers are technically-skilled hi-fi audio enthusiasts, I wrote an article on the fundamentals of interpolation as it's used to improve the performance of analog-to-digital conversion. Perhaps that article will be of some value to the subscribers of dsprelated.com. Here's what I wrote: We encounter the process of digital-to-analog conversion every day—in telephone calls (land lines and cell phones), telephone answering machines, CD & DVD players, iPhones, digital television, MP3 players, digital radio, and even talking greeting cards. This material is a brief tutorial on how sample rate conversion improves the quality of digital-to-analog conversion.