## Digital PLL's - Part 2

In 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 Swiss Army Knife of Digital Networks

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

## Digital PLL's -- Part 1

We will use Matlab to model the DPLL in the time and frequency domains (Simulink is also a good tool for modeling a DPLL in the time domain). Part 1 discusses the time domain model; the frequency domain model will be covered in Part 2. The frequency domain model will allow us to calculate the loop filter parameters to give the desired bandwidth and damping, but it is a linear model and cannot predict acquisition behavior. The time domain model can be made almost identical to the gate-level system, and as such, is able to model acquisition.

## 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

Recently 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

This 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

Elliptic 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

This 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

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.

## Complex Digital Signal Processing in Telecommunications

Digital 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.

## Digital Signal Processor Fundamentals and System Design

Digital 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.

## A Wide-Notch Comb Filter

This article describes a linear-phase comb filter having wider stopband notches than a traditional comb filter.

## A New Approach to Linear Filtering and Prediction Problems

In 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.

## Model Signal Impairments at Complex Baseband

In this article, we develop complex-baseband models for several signal impairments: interfering carrier, multipath, phase noise, and Gaussian noise. To provide concrete examples, we'll apply the impairments to a QAM system. The impairment models are Matlab functions that each use at most seven lines of code. Although our example system is QAM, the models can be used for any complex-baseband signal.

## Peak-to-Average Power Ratio and CCDF

Peak 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.

## Generating Complex Baseband and Analytic Bandpass Signals

There 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.

## Algorithms for Efficient Computation of Convolution

Convolution is an important mathematical tool in both ﬁelds of signal and image processing. It is employed in ﬁltering, denoising, edge detection, correlation, compression, deconvolution, simulation, and in many other applications. Although the concept of convolution is not new, the efﬁcient computation of convolution is still an open topic. As the amount of processed data is constantly increasing, there is considerable request for fast manipulation with huge data. Moreover, there is demand for fast algorithms which can exploit computational power of modern parallel architectures.

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

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

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

Endowing 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.