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**Jawaharlal Nehru Technological University Anantapur (JNTU) College of Engineering (CEP), Pulivendula, Pulivendula, Andhra Pradesh, India - JNTUACEP**- 19 Topics
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- Introduction - ( 2 - 2 )
- Sampling theory - ( 3 - 13 )
- Discrete-time signals - ( 14 - 25 )
- Discrete-time Systems - ( 26 - 37 )
- MATLAB (Linear Convolution) - ( 38 - 41 )
- Causality - ( 42 - 55 )
- Fourier analysis of discrete-time signals and systems - ( 56 - 78 )
- The z-transform and realization of digital filters - ( 79 - 159 )
- The Lattice structure- Introduction - ( 160 - 162 )
- Discrete Fourier series - ( 163 - 199 )
- Filtering through DFT/FFT - ( 200 - 215 )
- Fast Fourier transform - ( 216 - 232 )
- IIR Digital Filters - ( 233 - 258 )
- Digital filter design- The Butterworth filter - ( 259 - 287 )
- FIR digital filters - ( 288 - 369 )
- Multirate DSP - ( 370 - 390 )
- Up-sampling - ( 391 - 427 )
- FINITE WORD LENGTH EFFECTS - ( 428 - 428 )
- LIMIT CYCLES - ( 429 - 445 )

Topic:

Digital Signal Processing – 1 I. Introduction 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. Contents: Sampling theory Discrete-time signals Transformation of the independent variable Discrete-time systems Linear constant coefficient difference equations Fourier analysis of discrete-time signals and systems Frequency response of discrete-time system Properties of the discrete-time Fourier transform (DTFT) 1 of 77 2

Sampling theory Illustrative example A continuous-time random signal is shown. Based on this several important concepts are shown below. The signal is a continuous-time signal with continuous amplitude. Such a signal is also called an analog signal. x(t) + t 0 – 4 3 2 1 0 –1 –2 –3 –4 x(t) 8 7 6 5 4 3 2 1 0 nT 0 n 0 x(n) 5.5 5 Time 1T 1 2.8 2T 2 3.8 3T 3 5.3 4T 4 1.5 5T 5 4.6 6T 6 8.4 2 3 5 1 4 7 7T 7 6.9 8T 8 7.3 Time Sampled signal. Discrete-time signal – time is discrete, amplitude is continuous. Quantized. Quantization noise 6 7 (error). Digital signal – both time and amplitude are discrete. 111 110 111 Encoded to 3 bits/sample. Note this particular point exhibits saturation (out of range). Rounded down to 7, not 8. 101 010 011 101 001 100 If we were to represent every sample value with infinite precision (for example, x(1) = 2.8--, instead of being approximated as 2 or 3) then we would need registers and memory words of arbitrarily large size. However, owing to a finite word length we round off the sample values (in this case x(1) = 2.8-- will be rounded to 2). This introduces quantization noise or error. 2 of 77 3

The procedure of generating a discrete-time signal from an analog signal is shown in the following block diagram. In the digital signal processing course we are mostly dealing with discrete–time rather than digital signals and systems, the latter being a subset of the former. Sample (& Hold) Continuous-time, continuous amplitude (Analog signal) Quantizer Discrete-time, continuous amplitude Encoder Discrete-time, discrete amplitude (Digital signal) Encoded digital signal The three boxes shown above can be represented by an analog to digital converter (ADC). A complete digital signal processing (DSP) system consists of an ADC, a DSP algorithm (e.g., a difference equation) and a digital to analog converter (DAC) shown below. x(t) x(n) ADC Algorithm (Diff. Eq. or y(n) y(t) DAC equivalent) As the name implies discrete-time signals are defined only at discrete instants of time. Discrete-time signals can arise by sampling analog signals such as a voice signal or a temperature signal in telemetry. Discrete-time signals may also arise naturally, e.g., the number of cars sold on a specific day in a year, or the closing DJIA figure for a specific day of the year. AT&T’s T1 Stream The voice signal is band limited to 3.3 kHz, sampled at 8000 Hz (8000 samples per second), quantized and encoded into 8 bits per sample. Twenty four such voice channels are combined to form the T1 stream or signal. 1 = 0.125 msec. Sampling interval = 8000 Hz samples bits Bit rate for each channel = 8000 x8 = 64000 bits/sec. sec sample Bit rate for T1 = 64000 bits/sec per channel x 24 channels = 1 544 000 bits/sec. Commercial examples CD, Super Audio CD (SACD), DVD Audio (DVD-A), Digital audio broadcasting - 32 kHz, and Digital audio tape (DAT) - 48 kHz. Commercial Audio Examples CD Sampling Rate 44.1 kHz Super Audio CD (SACD) 2.8224 MHz Coding 16-bit PCM per sample With 2 channels the bit rate is 1.4112 Mbits/sec, but additional error control bits etc., raise it to 4.3218 Mbits/sec. 1-bit DSD (Direct Stream Digital – like Delta modulation) 3 of 77 4 DVD-Audio (DVD-A) 44.1, 88.2 or 48, 96, 192 kHz 12-, 20-, 24bit

Pulse-train sampling For pulse-train sampling of the signal x(t) by the rectangular pulse-train p(t) resulting in the sampled signal xs(t), we have xs(t) = x(t) p(t) p(t) The Fourier series of p(t) is given by p(t) = C e n j n 2 Fs t n , x(t) X where Fs = 1/T and the Fourier coefficients, are T/2 1 j n 2 Fs t Cn = dt p(t)e T T / 2 Thus xs(t) xs(t) = C n n x(t)e j n 2 Fs t The Fourier spectrum of xs(t) is given by Xs(F) = x (t)e j 2 F t s dt = C n x(t)e j n 2 Fs te j 2 F t dt Interchanging the order of integration and summation yields the aliasing formula C X (F nF ) dt = C x(t) e n s n j 2 ( F nFs ) t Xs(F) = n n = …+ C2 X (F 2Fs ) + C1 X (F 1Fs ) + C0 X (F)+ C1 X (F 1Fs ) + C2 X (F 2Fs ) +… x(t) t T 0 p(t) 2T 3T τ 1 t 0 T 2T 3T The sampled signal spectrum Xs(F) is sketched below. For convenience of illustration we have assumed the base band spectrum, X(F), to be real valued; the maximum value of |X(F)| is taken to be 1. Xs(F) consists of replicas of X(F), scaled by the Fourier coefficients Cn and repeated at intervals of Fs. Specifically, the replica at the origin is simply X(F) scaled by C0. Note that the magnitudes, |Cn|, have even symmetry. In this case, since FM ≤ Fs–FM, there is no overlap among the replicas in the graph of Xs(F). As a result the original signal x(t) can be 4 of 77 5

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