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Course: ECE 246/446 - Digital Signal Processing Meets: GOER 108, TR: 12:30 -1:45 Instructor: Jeff Astheimer Phone: 624-1582, 624-9778 Email: jeffasth@frontiernet.net Office: (ECE LIBRARY) Office Hours: Tuesday (11:30-12:30) and by appointment Teaching Assistant: Prerequisites: ECE241 and Matlab programming skills Books: Discrete - Time Signal Processing (3rd Edition) by Alan V. Oppenheim and Ronald W. Schafer Computer - Based Exercises for Signal Processing Using Matlab by James H. McClellan, et. al. Exams: Midterm (20%), Final Exam (20%) Homework: Assignments (30%) Matlab: Assignments (30%) Digital Signal Processing -Introduction- General Introduction (1) Signal implies the transmission of a message During transmission the message may undergo transformations that alters it. Radio signals are a good example. The original sounds experience a series of transformations as they make their way to the ears of the listener. General Introduction (2) An abbreviated sequence is : 1. Sound is converted into electrical signals 2. Electrical signals are converted into radio waves 3. Radio waves are reflected (multiple times) by layers of the ionosphere before reaching the receiving antenna 4. Radio waves are converted back into electricity 5. Electrical signals are transformed into sound General Introduction (3) These transformations are illustrated in the figure below : General Introduction (3) Corrupting influences may appear at each step Corruption may occur from background noise Or, the distortion may be more systematic, as in electronic circuits or components that have less than ideal characteristics. A diagram always helps! In a diagram we can make each step seem more manageable by representing it as a box. General Introduction (4) The transformations listed above may be represented in a diagram as follows : Sound to Voltage to Transmission of Radio waves Voltage to Voltage Radio Waves Radio Waves to Voltage sound Each box has an input and an output, and the original message makes it way from one end of the process to the other, undergoing each transformation in succession. An early goal of signal processing was to model these transformations and to add additional boxes to correct the distortions. The corrections (often called filters) were typically implemented as analog electronic circuits. General Introduction (5) Magnitude Magnitude Ultrasonic Imaging Provides another Example General Introduction (6) z0 Transmit at Recieve at positions x1 , x2 ,… positions y1 , y2 ,… Reflections to and from a single transducer are often not sufficient to form a good image, so signals from multiple transmitters and receivers must be combined General Introduction (7) In medical ultrasound the signals are distorted by tissues that are encountered en route to the objects being imaged General Introduction (8) Early conceptions of signal processing have evolved. The tools of the trade have been extended to higher dimensional data. Added complexities of higher dimensions don't appear, for example, in the transmission of television signals. Once an image has been rasterized it is essentially one dimensional and can be treated the same way as audio signals. General Introduction (9) But optical transmission that occurs in image formation is distinctly different. Since points of an image can be influenced by neighbors from every direction, local distortions can be much more complex. General Introduction (10) Sometimes interest is in the transformation itself (The Media is the Message), and the signals are only used used as probes as in X - rays. The ratio of the intensity of the output to the intensity of the input determines the density of the material that was encountered along each ray. General Introduction (11) Modern imaging methods make the old X - ray systems seem simplistic. Reconstruction of 2 - and 3 - dimensional facsimiles (via back -projection or Fourier inversions) are common, and these techniques have become part of the DSP arsenal. General Introduction (12) Physical principles that are being employed in sensors, detectors and transducers (e.g. MRI or PET imaging, etc.) are becoming more and more sophisticated Part of the interest and challenge in DSP is to blend the processing theory with the physical theory of the measurement. In experimental settings things always goes wrong, and you need to figure out if the problem is in the physics or in the processing. General Introduction (13) Another common DSP task is identification of special features in a signal. The feature could be as simple as a peak or a voltage transition, but could also be more complicated, as in character or pattern recognition applications. This image shows time varying potentials at different electodes around the head. When the subject is presented with a stimulus, a potential change is evoked at different times for different electrodes. The signals must be processed to identify the evoked potentials General Introduction (14) We have been discussing 'Signal Processing', What about the word 'Digital'? x ( nT ) x (t ) x ( ( n + 1) T ) T 'Digital' indicates concern with signals that are made up of discretely digitized samples. Digitization is a critical practical matter. Signal processing depends on digital electronics. General Introduction (15) As signals have proliferated, new processing steps have have been mandated by practical considerations. This chart shows FCC frequency band allocations for the electomagnetic spectrum General Introduction (16) Various bands fill up with telephone conversations, cartoon shows, talk radio etc., making it necessary to find ways to encode voice and other signals into condensed forms . Thus, compression and decompression algorithms have become very important Compress Transmit Decompress Compressed Transmission Also, to maintain some semblance of privacy, encryption and decryption coding has become essential. Encrypt Transmit Decrypt Encrypted Transmission Characterization of DSP (1) These examples are only a small sampling of a very wide range of areas that digital signal processing has branched into. However, the fundamental principles are well established, and have evolved into an elegant theory. The next few slides try to identify some of the distinctive characteristics of this theory. Characterization of DSP (2) To begin with, DSP procedures are operations that are applied to discrete signals. However, sampling of continuous (analog) signals (A/D Conversion), and also interpolation of discrete signals into continuous forms (D/A Conversion), enable the discrete operations to have a broader range of application. Continuous A/D Digital Signal D/A Continuous Input Converter Processing Converter Output Continuous Signal Processing Characterization of DSP (3) The purposes of the DSP operations are varied. Some of the more important ones are • Correct Distortion (de - blurr) • Signal Decomposition (separate messages or separate message and noise) • Feature Enhancement (boost signal components, sharpen images, etc.) • Noise Reduction (classical smoothing) • Signal Analysis (transitions, patterns, peaks, frequency distribution, etc.) • Signal Compression • Signal Encryption • Signal Transformation (Fourier, Walsh, Wavelet) Characterization of DSP (4) It may seem like almost any type of processing can be placed under the DSP heading, but this isn't so. Many operations (e.g statistical calculations such as the analysis of variance) belong to altogether different disciplines. • DSP procedures are distinguished by being closely tied to the operations of convolution and Fourier decomposition. That an entire discipline is devoted to such a seemingly limited set of operations results from a confluence of three factors : Characterization of DSP (5) Factor 1. A very large number of naturally occuring and man made signal influences share two important properties : 1. The influences are time invariant 2. The influences are linear (satisfy a superposition principle) These types of influences are called linear time invariant systems (or simply LTI systems) and are characterized by convolution. Characterization of DSP (6) Factor 2. Analysis of convolution operators is greatly facilitated by Fourier (harmonic) decomposition, for which there is a very large theoretical foundation. The theory of Fourier analysis is at the heart of many scientific disciplines (linear differential equations, potential theory, heat conduction, wave propagation, etc.) Characterization of DSP (7) Factor 2. (continued) Many disciplines contribute alternate models that can help to understand and interpret different types of LTI systems. Broad interest in these problems has stimulated development of the subject. Characterization of DSP (8) Factor 3. Despite the preceding factors, DSP would probably not play such an important role in engineering were it not for the fact that the principal processing step in all DSP operations is a simple : multiply and add. This feature is immediately apparent in the convolution operation, and can be easily verified for the discrete Fourier transform as well. Characterization of DSP (9) Factor 3. (continued) Furthermore, the Fast Fourier Transform, which was rediscovered by Cooley and Tukey in the 60's, recasts the transform calculation as an alterante set of multiply and add operations that is extremely efficient. The ability of digital hardware, in the form of either a computer or a dedicated signal processing component (DSP chip, array processor, etc.) to perform massive repititions of the multiply and add operation (either serially or in parallel) means that DSP can be brought to bear on signals in all kinds of circumstances and, in particular, in real time. This is what makes it possible for digital signal processing to be used in place of the older and cruder approach of using analog cicuits. Characterization of DSP (10) This last factor gives DSP a rather different flavor from more a straightforward study of Fourier analysis, or studies of mathematical models in other scientific disciplines. For although digital hardware may seem to have unlimited capability, there are always limits, and to accomodate these limits the subject is infused with a practical aspect. A standard engineering question is not 'How can this filter be implemented ?' but rather, 'How can this filter be implemented using only 15 multiply and adds for each input sample?' Course Outline (1) The text for the course is Discrete - Time Signal Processing (3rd Edition) by Alan V. Oppenheim and Ronald W. Schafer We will try to cover the first 8 chapters. Course Outline (2) Chapter 2 : Discrete - Time Signals and Systems • Discrete - time sequences are introduced and their basic operations and features are described • The notion of a system as an operator is explained and LTI characteristics are distinguished. • LTI systems can sometimes be characterized a solutions to a linear constant coefficient difference equation (LCCDE). This characterization will have special importance for the practical implementation of real time systems. • The Discrete - Time Fourier Transform is defined and important properties of the transform are identified Course Outline (3) Chapter 3 : The z - transform • The z - transform is an extension of the discrete - time Fourier transform to regions of the complex plain. The importance of this transform won't be fully appreciated until chapter 5. • Important properties of the transform are identified. • Methods of computing the z - transform are discussed, with special emphasis placed on determining the z - transform of associated with LCCDE's Course Outline (4) Chapter 4 : Sampling of Continuous - Time Signals • This chapter examines A/D and D/A transformations, and characterizes these operations in both the time and frequency domain. This leads to the fundamental notion of the nyquist sampling rate. • The D/A operation provides a natural way to associate a continuous signal to any discrete sequence, and this association is often exploited to apply continuous operations into discrete sequences. • An example of particular importance is the operation of changing the sampling rate of a discrete - time signal. Course Outline (5) Chapter 5 : Transform Analysis of LTI systems • The best way to understand a LTI system is to study the frequency response function of the system (the Fourier transform of the impulse response). The chapter begins with a discussion of some basic characteristics of the frequency response function. • The role of the z - transform is not to help understand systems, but rather to help design them. The placement of poles and zeros of the z - transform of the impulse response of LCCDE systems dictates the behavior of the frequency response. • Special configurations of poles and zeros lead to systems with special properties : stable, causal, all -pass, minimum -phase, linear phase Course Outline (6) Chapter 6 : Discrete Time Structures • These structures refer to different schemes for implementing linear systems that derive from LCCDE's. The purpose of the structures is to enable systematic exploration of different implementations to optimize their efficiency and numerical precision. • Diagrammatic representation of different implementations are described and categorized. • Methods for determining the numerical error associated with different diagrams are explored. Course Outline (7) Chapter 7 : Filter Design Techniques • A filter is just a LTI system with special frequency response characteristics. But real time applications often require the filter to be implemented as an LCCDE system, which means that it may not be possible to attain the ideal (i.e. exact) response. • Filter design entails choosing the LCCDE coefficients to approximate the frequency response in an optimal manner Course Outline (8) Chapter 8 : The Discrete Fourier Transform • The Discrete Fourier transform is distinct from the Discrete - Time Fourier transform described in chapter 2 because it only applies to discrete sequences that are periodic. • The effect of this simplification is that the frequency response function also becomes a periodic discrete sequence. • This has far reaching effects on prior methods, and offers a powerful alternative approach to system analysis and design, but this alternative is not as amenable to real time applications. Homework • Collaboration : Working together has definite benefits and can make homework assignments much more palatable. However, it muddles the significance of the homework grade, and can diminish the efficacy of homework as a learning device. The recommended policy is to only collaborate judiciously. • Clarity : An important part of the grading for both homework and matlab assignments will be clarity of presentation. This means, in particular, that matlab code should be well organized and well documented (lots of comments)

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