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.
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 :-).
Method to Calculate the Inverse of a Complex Matrix using Real Matrix Inversion
This paper describes a simple method to calculate the invers of a complex matrix. The key element of the method is to use a matrix inversion, which is available and optimised for real numbers. Some actual libraries used for digital signal processing only provide highly optimised methods to calculate the inverse of a real matrix, whereas no solution for complex matrices are available, like in [1]. The presented algorithm is very easy to implement, while still much more efficient than for example the method presented in [2]. [1] Visual DSP++ 4.0 C/C++ Compiler and Library Manual for TigerSHARC Processors; Analog Devices; 2005. [2] W. Press, S.A. Teukolsky, W.T. Vetterling, B.R. Flannery; Numerical Recipes in C++, The art of scientific computing, Second Edition; p52 : “Complex Systems of Equations”;Cambridge University Press 2002.
Design of a Scalable Polyphony-MIDI Synthesizer for a Low Cost DSP
In this thesis, the design of a music synthesizer implementing the Scalable Polyphony-MIDI soundset on a low cost DSP system is presented. First, the SP-MIDI standard and the target DSP platform are presented followed by review of commonly used synthesis techniques and their applicability to systems with limited computational and memory resources. Next, various oscillator and filter algorithms used in digital subtractive synthesis are reviewed in detail. Special attention is given to the aliasing problem caused by discontinuities in classical waveforms, such as sawtooth and pulse waves and existing methods for bandlimited waveform synthesis are presented. This is followed by review of established structures for computationally efficient time-varying filters. A novel digital structure is presented that decouples the cutoff and resonance controls. The new structure is based on the analog Korg MS-20 lowpass filter and is computationally very efficient and well suited for implementation on low bitdepth architectures. Finally, implementation issues are discussed with emphasis on the Differentiated Parabole Wave oscillator and MS-20 filter structures and the effects of limited computational capability and low bitdepth. This is followed by designs for several example instruments.
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.
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
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.
EngD thesis: Reduced-Complexity Signal Detection in Digital Communications Receivers
The Author began this Engineering Doctorate (EngD) in Autumn 1999, whilst already in full-time employment as a DSP software engineer at Nortel Networks, Harlow. This EngD comprises a set of three projects. The first project was focused on the development of dual-tone multi-frequency (DTMF) signal detection software. DTMF signals are currently used for operating menu-driven services such as voice-mail, telephone banking and share-dealing. The need for detection software in a packet networking environment exists because such signals become degraded when they travel through speech compression circuits. In addition, if they appear as echoes on a telephone line, they can affect the operation of echo cancellation systems. In this thesis a number of DSP algorithms are discussed where fast detection and minimum complexity are key characteristics. A key contribution here was the development of a novel detection algorithm based on the extraction of parameters that model the DTMF signal. The thesis reports a method combining parameter extraction with the technique of maximum likelihood to perform DTMF detection, resulting in very short time-frames when compared to standard approaches. Reducing the complexity of detection techniques is also important in today’s communication systems. To this end a key contribution here was the development of the ‘split Goertzel algorithm’, which permitted an overlapping of analysis windows without the need for reprocessing input data. Besides being applied to voice-band signals, such as in the case of DTMF, the Author also had the opportunity during the EngD to apply the skills and knowledge acquired in signal processing to higher-rate data-streams. This involved work concerning the equalisation of channel distortion commonly found in wireless communication systems. This covers two projects, with the first being conducted at Verticalband Ltd. (now no longer operational) in the area of digital television receivers. In this part of the thesis a real-time DSP implementation is discussed for enhancing a simulation system developed for wireless channels. A number of channel equalisation approaches are studied. The work concludes that for high-rate signals, non-linear algorithms have the best error rate performance. On the basis of the studies carried out, the thesis considers development and implementation issues of designs based on the decision feedback equaliser. The thesis reports on a software design which applies the method of least squares to carry out filter coefficient adaptation. The Verticalband studies reported lead on to further research within the context of channel equalisation, in the context of the detection of data in multiple-input multiple-output (MIMO) wireless local area network (WLAN) systems. This work was undertaken at Philips Research in Eindhoven, The Netherlands. The thesis discusses implementation scenarios of multi-element antenna arrays that aim to provide bit-rates orders of magnitude higher than today’s WLAN offerings, as those required by emerging standards such as 802.11n. The complexity of optimal detection techniques, such as maximum likelihood, scales exponentially with the number of transmit antennas. This translates to high processing costs and power consumption, rendering such techniques unsuitable for use in battery-powered devices. The initial main contribution here was the analysis of the complexity of algorithms whose performance had already been tested, such as the non-linear maximum likelihood approach and also less complex methods utilising linear filtering. This resulted in the development of new formulae to predict processing costs of algorithms based on the number of transmit and receive antennas. Other key contributions were defining a method to reduce the complexity of matrix inversion when using the Moore-Penrose pseudo-inverse, and the application of matrix decomposition to obviate the need for costly matrix inversion at all. Some on-going research into sub-optimal detection is also discussed, which describes methods to reduce the search-space for the maximum likelihood algorithm.
Acoustic Echo Cancellation using Digital Signal Processing
Acoustic echo cancellation is a common occurrence in todays telecommunication systems. It occurs when an audio source and sink operate in full duplex mode, an example of this is a hands-free loudspeaker telephone. In this situation the received signal is output through the telephone loudspeaker (audio source), this audio signal is then reverberated through the physical environment and picked up by the systems microphone (audio sink). The effect is the return to the distant user of time delayed and attenuated images of their original speech signal. The signal interference caused by acoustic echo is distracting to both users and causes a reduction in the quality of the communication. This thesis focuses on the use of adaptive filtering techniques to reduce this unwanted echo, thus increasing communication quality. Adaptive filters are a class of filters that iteratively alter their parameters in order to minimise a function of the difference between a desired target output and their output. In the case of acoustic echo in telecommunications, the optimal output is an echoed signal that accurately emulates the unwanted echo signal. This is then used to negate the echo in the return signal. The better the adaptive filter emulates this echo, the more successful the cancellation will be. This thesis examines various techniques and algorithms of adaptive filtering, employing discrete signal processing in MATLAB. Also a real-time implementation of an adaptive echo cancellation system has been developed using the Texas Instruments TMS320C6711 DSP development kit.
Audio Time-Scale Modification
Audio time-scale modification is an audio effect that alters the duration of an audio signal without affecting its perceived local pitch and timbral characteristics. There are two broad categories of time-scale modification algorithms, time-domain and frequency-domain. The computationally efficient time-domain techniques produce high quality results for single pitched signals such as speech, but do not cope well with more complex signals such as polyphonic music. The less efficient frequencydomain techniques have proven to be more robust and produce high quality results for a variety of signals; however they introduce a reverberant artefact into the output. This dissertation focuses on incorporating aspects of time-domain techniques into frequency-domain techniques in an attempt to reduce the presence of the reverberant artefact and improve upon computational demands. From a review of prior work it was found that there are a number of time-domain algorithms available and that the choice of algorithm parameters varies considerably in the literature. This finding prompted an investigation into the effects of the choice of parameters and a comparison of the various techniques employed in terms of computational requirements and output quality. The investigation resulted in the derivation of an efficient and flexible parameter set for use within time-domain implementations. Of the available frequency-domain approaches the phase vocoder and timedomain/ subband techniques offer an efficiency and robustness advantage over sinusoidal modelling and iterative phase update techniques, and as such were identified as suitable candidates for the provision of a framework for further investigation. Following from this observation, improvements in the quality produced by time-domain/subband techniques are realised through the use of a bark based subband partitioning approach and effective subband synchronisation techniques. In addition, computational and output quality improvements within a phase vocoder implementation are achieved by taking advantage of a certain level of flexibility in the choice of phase within such an implementation. The phase flexibility established is used to push or pull phase values into a phase coherent state. Further improvements are realised by incorporating features of time-domain algorithms into the system in order to provide a ‘good’ initial set of phase estimates; the transition to ‘perfect’ phase coherence is significantly reduced through this scheme, thereby improving the overall output quality produced. The result is a robust and efficient time-scale modification algorithm which draws upon various aspects of a number of general approaches to time-scale modification.
Voice Activity Detection. Fundamentals and Speech Recognition System Robustness
An important drawback affecting most of the speech processing systems is the environmental noise and its harmful effect on the system performance. Examples of such systems are the new wireless communications voice services or digital hearing aid devices. In speech recognition, there are still technical barriers inhibiting such systems from meeting the demands of modern applications. Numerous noise reduction techniques have been developed to palliate the effect of the noise on the system performance and often require an estimate of the noise statistics obtained by means of a precise voice activity detector (VAD). Speech/non-speech detection is an unsolved problem in speech processing and affects numerous applications including robust speech recognition, discontinuous transmission, real-time speech transmission on the Internet or combined noise reduction and echo cancellation schemes in the context of telephony. The speech/non-speech classification task is not as trivial as it appears, and most of the VAD algorithms fail when the level of background noise increases. During the last decade, numerous researchers have developed different strategies for detecting speech on a noisy signal and have evaluated the influence of the VAD effectiveness on the performance of speech processing systems. Most of the approaches have focussed on the development of robust algorithms with special attention being paid to the derivation and study of noise robust features and decision rules. The different VAD methods include those based on energy thresholds, pitch detection, spectrum analysis, zero-crossing rate, periodicity measure, higher order statistics in the LPC residual domain or combinations of different features. This chapter shows a comprehensive approximation to the main challenges in voice activity detection, the different solutions that have been reported in a complete review of the state of the art and the evaluation frameworks that are normally used. The application of VADs for speech coding, speech enhancement and robust speech recognition systems is shown and discussed. Three different VAD methods are described and compared to standardized and recently reported strategies by assessing the speech/non-speech discrimination accuracy and the robustness of speech recognition systems.
