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Note for Digital Signal Processing - DSP By Sreedhar Nagavelly

  • Digital Signal Processing - DSP
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UNIT I – INTRODUCTION Syllabus Introduction to Digital Signal Processing: Discrete time signals & sequences, linear shift invariant systems, stability, and causality. Linear constant coefficient difference equations. Frequency domain representation of discrete time signals and systems. Introduction Signal A signal is any physical quantity that carries information, and that varies with time, space, or any other independent variable or variables. Mathematically, a signal is defined as a function of one or more independent variables. 1 – Dimensional signals mostly have time as the independent variable. For example, Eg., S1 (t) = 20 t2 2 – Dimensional signals have two independent variables. For example, image is a 2 – D signal whose independent variables are the two spatial coordinates (x,y) Eg., S2 (t) = 3x + 2xy + 10y2 Video is a 3 – dimensional signal whose independent variables are the two spatial coordinates, (x,y) and time (t). Similarly, a 3 – D picture is also a 3 – D signal whose independent variables are the three spatial coordinates (x,y,z). Signals S1 (t) and S2 (t) belong to a class that are precisely defined by specifying the functional dependence on the independent variables. Natural signals like speech signal, ECG, EEG, images, videos, etc. belong to the class which cannot be described functionally by mathematical expressions. System A system is a physical device that performs an operation on a signal. For example, natural signals are generated by a system that responds to a stimulus or force. For eg., speech signals are generated by forcing air through the vocal cords. Here, the vocal cord and the vocal tract constitute the system (also called the vocal cavity). The air is the stimulus. The stimulus along with the system is called a signal source. An electronic filter is also a system. Here, the system performs an operation on the signal, which has the effect of reducing the noise and interference from the desired information – bearing signal. When the signal is passed through a system, the signal is said to have been processed. Processing The operation performed on the signal by the system is called Signal Processing. The system is characterized by the type of operation that it performs on the signal. For example, if the operation is linear, the system is called linear system, and so on. GVPW DIGITAL SIGNAL PROCESSING Page 2

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Digital Signal Processing Digital Signal Processing of signals may consist of a number of mathematical operations as specified by a software program, in which case, the program represents an implementation of the system in software. Alternatively, digital processing of signals may also be performed by digital hardware (logic circuits). So, a digital system can be implemented as a combination of digital hardware and software, each of which performs its own set of specified operations. Basic elements of a Digital Signal Processing System Most of the signals encountered in real world are analog in nature .i.e., the signal value and the independent variable take on values in a continuous range. Such signals may be processed directly by appropriate analog systems, in which case, the processing is called analog signal processing. Here, both the input and output signals are in analog form. These analog signals can also be processed digitally, in which case, there is a need for an interface between the analog signal and the Digital Signal Processor. This interface is called the Analog – to – Digital Converter (ADC), whose output is a digital signal that is appropriate as an input to the digital processor. In applications such as speech communications, that require the digital output of the digital signal processor to be given to the user in analog form, another interface from digital domain to analog domain is required. This interface is called the Digital – to – Analog Converter (DAC). In applications like radar signal processing, the information extracted from the radar signal, such as the position of the aircraft and its speed are required in digital format. So, there is no need for a DAC in this case. Block Diagram Representation of Digital Signal Processing analog input signal Analog - to Digital Converter (ADC) Digital Signal Processor (DSP) Digital - to Analog Converter (DAC) Analog output signal Advantages of Digital Signal Processing over Analog Signal Processing 1. A digital programmable system allows flexibility in reconfiguring the digital signal processing operations simply by changing the program. Reconfiguration of an analog system usually implies a redesign of the hardware followed by testing and verification. 2. Tolerances in analog circuit components and power supply make it extremely difficult to control the accuracy of analog signal processor. A digital signal processor provides better control of accuracy requirements in terms of word length, floating – point versus fixed – point arithmetic, and similar factors. 3. Digital signals are easily stored on magnetic tapes and disks without deterioration or loss of signal fidelity beyond that introduced in A/D conversion. So the signals become transportable and can be processed offline. 4. Digital signal processing is cheaper than its analog counterpart. 5. Digital circuits are amenable for full integration. This is not possible for analog circuits because inductances of respectable value (μH or mH) require large space to generate flux. 6. The same digital signal processor can be used to perform two operations by time multiplexing, since digital signals are defined only at finite number of time instants. GVPW DIGITAL SIGNAL PROCESSING Page 3

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7. Different parts of digital signal processor can work at different sampling rates. 8. It is very difficult to perform precise mathematical operations on signals in analog form but these operations can be routinely implemented on a digital computer using software. 9. Several filters need several boards in analog signal processing, whereas in digital signal processing, same DSP processor is used for many filters. Disadvantages of Digital Signal Processing over Analog Signal Processing 1. Digital signal processors have increased complexity. 2. Signals having extremely wide bandwidths require fast – sampling – rate ADCs. Hence the frequency range of operation of DSPs is limited by the speed of ADC. 3. In analog signal processor, passive elements are used, which dissipate very less power. In digital signal processor, active elements like transistors are used, which dissipate more power. The above are some of the advantages and disadvantages of digital signal processing over analog signal processing. Discrete – time signals A discrete time signal is a function of an independent variable that is an integer, and is represented by x [ n ] , where n represents the sample number (and not the time at which the sample occurs). A discrete time signal is not defined at instants between two successive samples, or in other words, for non – integer values of n. (But, it is not zero, if n is not an integer). Discrete time signal representation The different representations of a discrete time signal are 1. Graphical Representation Graphic al Representation 4 3 DT signal x[n] 2 1 0 -1 -2 -3 -4 -3 -2 -1 0 1 sample number n 2 3 4 2. Functional representation 1, 𝑓𝑜𝑟 𝑛 = 1, 2 3 𝑥[𝑛] = { 4, 𝑓𝑜𝑟 𝑛 = 2 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 3. Tabular representation N - - - - - - -2 x[n] - - - - - 0 GVPW -1 0 0 1 1 1 2 4 3 1 4 0 DIGITAL SIGNAL PROCESSING 5 0 - - - - - - - - - - - Page 4

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