Asymmetric windows in digital signal processing

Abstract

Symmetric windows are widely used in the field of digital signal processing due to their easy design and linear phase property. Nevertheless, symmetry also implies a few potential drawbacks like longer time delay in short-time frequency analysis and some limitations in frequency response. The removal of the symmetry constraint can therefore lead to asymmetric windows better in certain respects. In signal processing, better signal representations and related improved processing performance can be accomplished. In addition, shorter time delay can be achieved with asymmetric windows. This feature is important for contemporary spoken communications in the Internet or mobile networks and all other real-time signal processing applications.

The article gives a comprehensive review of the past and current work in the field of asymmetric windows. We elaborate on our work and related efforts of other researchers inspired by the idea of asymmetry. Shorter time delay and some better spectral properties are the most prominent potential of asymmetric windows. However, there are also some other more subtle properties which can improve the performance in specific application contexts (e.g., frequency estimation and detection of closely spaced components in frequency analysis). Several examples of interesting effects of asymmetric windows are presented, followed by empirical evaluations in the fields of pitch modification, shorter time delay audio processing (e.g., speech coding), frequency analysis, speech processing, and FIR filter design. In addition, a detailed comparison of various asymmetric windows found in the literature to widely known symmetric windows is made taking into account several practical and theoretical aspects. Finally, all presented achievements are summarized in a table which provides a complete overview of the current state of this interesting research and application field.

Introduction

Digital signals are becoming a ubiquitous part of our life. Signal processing platforms are more and more compact, low-priced and capable of performing complex computing operations. Nowadays, we acquire and process many signals which were some years ago either not available or too complex to be processed. We are able to solve complex problems with low-priced desktop systems, and to produce small ubiquitous devices which can interact with the environment, other devices, and users via signals and other communication channels.

Despite the ongoing evolution, some concepts stay more or less constant and unchanged in the field of Digital Signal Processing (DSP). Signals have to be acquired through an analog-to-digital conversion process denoted as sampling, after which signals are processed in a digital form as a series of samples in the time domain or in the transformed domain (mostly frequency domain).

When processing digital signals, two tasks are generally performed. The first one is the information extraction task (e.g., speech recognition, frequency analysis) which tries to gain specific information from the signal's content. The second task is the transformation of the signal into a new form which is in certain aspects better than the original one (e.g., filtering, speech enhancement, denoising). Sometimes the time domain representation of the signal suffices for both tasks. However, in the majority of nontrivial cases and more complex signal processing procedures, the signal is transformed into some other domain (mostly frequency domain). Here the desired properties are better recognized and distinguished from other less important ones.

By far the most commonly used concept for the transformation of digital signals into the frequency domain is the Discrete Fourier Transform (DFT), which reveals the frequency content of the signal more informative for a series of signal processing tasks. We denote such procedure of determining the frequency content of a signal as frequency analysis and we will put a lot of focus on it in this article. It is particularly important because it serves as a starting point for various methods from the fields of speech, image and audio processing. Those are among the most widespread research and practical application areas of DSP.

Several examples of such applications will be presented in this article. They will demonstrate how improvements in frequency analysis can enhance the performance of systems which build on the time-frequency presentation of the signals. As we have said, there are many such systems which could easily benefit from the improved results of frequency analysis. Therefore, we could achieve multiplicative effects on the widespread area of related techniques and systems. This is the main reason to acknowledge the high importance of frequency analysis as an elementary procedure in these fields.

Regarding the variability of a signal's frequency content, we can classify signals into two common groups. If a signal's frequency content is stationary, then it is commonly analyzed in a single segment with a finite number of samples. In this simple case, we do not have any time dimension in frequency presentation; it is mostly not needed anyway.

If the signal's content is nonstationary over time, then it is generally analyzed in a series of finite, overlapped segments (commonly denoted as frames) with the assumption of the stationarity in each frame (Fig. 2). We denote this procedure as a short-time frequency analysis which usually results in common three-dimensional (time, frequency, magnitude) signal representation denoted as a spectrogram (Fig. 3). In both cases, the “average” frequency content of the signal is calculated in each frame. In the second case of multiple frames, those are generally shorter and stationarity is assumed for a single frame only. Frame length is frequently determined in the context of the application, and is generally a compromise between the desired time and frequency resolutions of the signal's representation.

In both variations of frequency analysis, we perform analysis or processing tasks on frames with a finite number of samples. Samples outside of a particular frame are assumed to be equal to zero. Theoretically, this step is equivalent to a product between the infinite discrete signal being sampled and another infinite sequence of samples with non-zero values only inside the same frame and all zero values elsewhere. This sequence is denoted as a window and is a built-in part of any frame-based processing procedure on digital signals.

The emerging digital evolution has generated more and more powerful computing platforms available for digital signal processing tasks. However, there are two permanent aspects which require proper attention even in modern signal processing systems: the processing time delay and the undesired variability of signals. No matter how powerful and efficient our systems are, we always aspire to perform better, faster, and do more complex computations in the same time unit. Consequently, the two aspects will probably stay in focus in the future as well.

In this section, we further elaborate on the two highlighted aspects (time delay in processing and undesired signal variability) in separate subsections. We also present the concept of asymmetric windows as a potential approach to deal with negative effects of those aspects. In the last subsection, a few important general thoughts are highlighted, explaining the research path we follow in our work.

The rest of the chapter is organized as follows. Section 2 is a general overview of windows in signal processing with explanation of their influence in both approaches to frequency analysis and the Finite Impulse Response (FIR) filter design. A lot of practical applications in the signal processing area are based on these elementary tasks. Section 3 represents a more detailed introduction to asymmetric windows, their potentials, and design methods. The following section, which is the most important part of the article, gives a general review of related work on asymmetric windows. Various approaches to the design of asymmetric windows from various application areas are here presented and analyzed in more detail with the aim of evaluating the effect of presented asymmetric windows and explaining their advantages and possible pitfalls. Section 5 contains discussion and conclusions. The presented asymmetric windows and their utilization in various DSP application areas are summarized in a table. In addition, we identify the related promising research topics for the future.

In practice, we are only able to process signals using finite sequences of samples, which means dividing longer signals into shorter frames. Such approach is particularly important for real-time performance. We process signals in finite frames, on the one-by-one basis, and perform information extraction and/or signal transformation tasks on each frame separately. Consequently, it takes time to acquire all samples in the frame and carry out additional processing steps. Therefore, we cannot avoid the intrinsic time delay between the input signal and output results. We usually want to make this time delay as short as possible as most real-time signal processing applications severely depend on a short processing time delay.

Besides traditional signal processing tasks which depend on a short time delay, such as analysis, coding, distributed processing, recognition and communications, the time delay is also important for several emerging, interactive and user-centered tasks such as [1]:

• Hearing aids

• Augmented reality

• Computer-aided music practicing tools

• Brain-computer interface

• Patient monitoring systems

• Musical pitch tracking

• Real-time onset detection

For all the mentioned applications, the short time delay of processing is crucial for their performance. For most others, it is also a desired feature. Therefore, the requirement for a short processing time delay is basically built into almost every real-time application in the field of DSP.

Another important aspect of modern signal processing paradigms is the variability of signals. From the pure signal processing viewpoint, an ideal signal would be constant and without any noise. However, the signal's variability normally contains useful information and related variability is usually denoted as a desired variability. On the other hand, quite often there are various unwanted distortions present in signals, which are denoted as undesired variability. In practice, we always try to distinguish between those two types of variability in signals. This is usually a complex task, at least for nontrivial problems.

Generally, we want to focus on useful variability in signals. However, undesired variability commonly interferes with its useful counterpart in different ways. Therefore, we create signal presentations which merge both domains. In short-time frequency analysis, we denote such representation as Time-Frequency (TF). One axis represents time and the other frequency. We try to utilize both dimensions when focusing only on useful variability in signals.

The most common interfering variability is noise. It is always present in real signals at various power levels. The relation of power between the useful part of the signal and the unwanted noise is denoted as the Signal-to-Noise Ratio (SNR). However, the measure itself does not help much without the processing techniques which enhance signals achieving higher SNR ratios. In some cases, SNR loss is inevitable due to the processing of signals, and we only try to minimize that SNR loss; this situation is denoted a passive approach. On the other side, if we succeed to increase the SNR ratio (e.g., filtering, speech enhancement, denoising), those methods are usually recognized as an active approach.

The solution to these two major issues of modern signal processing (namely, time delay and variability of signals) seems easy at first sight. We simply need to design and implement systems which will perform with a short time delay and will be able to distinguish between the useful and the undesired variability in signals. Unfortunately, this is easier to describe in theory than achieve in practice. Still, there are various ways to improve existing systems in both directions.

One possibility is the use of asymmetric windows which can help tackle the stated problems in practice. As we will show in the continuation of the article, asymmetric windows can in certain respects be better than the commonly used symmetric windows. They can produce better signal presentations with a shorter time delay in processing. Despite being considered a passive approach, their integration is usually a simple replacement with no additional computing cost.

As already stated, asymmetric windows can have better spectral and time delay properties compared to the commonly used symmetric windows. As a simple, but a built-in step in a frame-based frequency analysis procedure, the influence on final results is generally expected not to be very significant. Since asymmetric windowing can be considered as a passive method, it tries to minimize the loss of the SNR ratio during frequency analysis. However, as we will show in the article, it can contribute to better results in most procedures based on frequency analysis. This fact represents the main motivation for this article.

There is some more interesting potential for the enhancement of frequency estimation results. Certain properties of asymmetric windows which result in better frequency representation could also contribute to a better performance of entire systems based on frequency analysis. In addition, we could empirically evaluate the hypothesis that passive asymmetric window improvement to a certain level also has an effect on the performance of complex systems which already operate in real conditions using symmetric windows. Some of our preliminary tests, carried out in a research work related to speech recognition, confirmed that exciting finding [2].

Finally, we would like to highlight some general thoughts on the evolution of the signal processing research area in the next subsection.

The first thought is the dilemma whether it is better to enhance the performance of systems by incorporating more and more knowledge or to rely on emerging machine learning concepts (e.g., deep learning models). The latter can learn from data even without initial domain knowledge and are capable of quite an outstanding performance. However, they also exhibit certain shortcomings (e.g., limited ability of explanation or presentation of gained knowledge).

During our research work, we have tested both approaches. However, we have always preferred the knowledge-based approach with proper empirical evaluation. We have focused on existing knowledge and constantly tried to expand it with actual research achievements. Consequently, we have always wanted to have a full understanding of a common workflow from the input, processing and output of various systems.

On the other hand, we feel that deep learning networks are a promising concept and deserve proper research attention. In addition, we can gain a lot of new knowledge from such systems. However, we want to foster knowledge-based approaches to maintain a proper level of understanding of the systems which perform more and more complex tasks. We also believe that we can enhance the performance of knowledge-based systems by inclusion of knowledge gained from machine learning concepts.

The second dilemma is related to the importance of system robustness to signal distortions not present in the learning phase. In our work, we denote this property as “inherent robustness.” This situation is quite common in many cases. Systems are usually designed and/or trained in “learning” conditions and it is practically impossible to include all “real” environment conditions in a system's learning phase. Therefore, systems' performance is often substantially degraded in a “real” environment. Actually, the systems which are able to perform successfully in different conditions are very rare. This is particularly true in the fields of speech processing and recognition. Consequently, we strongly believe that it is always important to enhance the system's inherent robustness and get better performance in this way although, this is often much harder and takes more time. However, if we perform this task successfully, even the systems which are trained in “laboratory” (clean) conditions can perform better in unseen real environmental conditions. In our opinion, this is the key point for further development of such systems and deserves a proper research attention.

As we have explained in the above subsections, we believe that asymmetric windows are a promising concept which needs proper research attention and can generate a lot of interesting new research ideas [3]. However, their implementations in signal processing systems require background knowledge in computer architecture and organization areas. Therefore, a significant effort is also needed in computer engineering education [4]. Both arguments contribute to the motivation for the work in the field of asymmetric windows presented in this article.