113_1_113-midrev2010-toclass

113_1_113-midrev2010-toclass - Why Digital Signal...

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1 Why Digital Signal Processing? Most signals are CT (e.g., speech and audio). But: • digital computers are fast (real time in many apps), efficient, and flexible and • Several techniques such as encryption (for security) and compression techniques (for reducing data rate) are mostly designed for digital data representations. • In this class, we dealt only with discrete- time signal processing Review *Discrete sequences, functions *Notions of Linearity, Time Invariance, Relaxed, Causality, BIBO, Periodic, Static/Dynamic (memory), Symmetry, Energy and Power Signals *Solving the output for systems characterized by difference equations; Block Diagrams * Filters (FIR, IIR), minimum phase * The Z-transform Material: lecture notes, Chs. 1, 2, 3, 4, 5, 6, 9, 10, and 11. and Homeworks 1-4. One 8.5 x 11 sheet (double-sided) is allowed. Nothing electronic (phones, calculators, etc.) Operations on signals • Discrete time signal often obtained by sampling a continuous-time signal • Sequence { x [ n ]} = x a ( nTs ), n =…-1,0,1,2… Ts = sampling period; 1/ T = Fs, where Fs is sampling frequency
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2 Sampling • Fs (sampling frequency or rate –SR-) needs to be at least twice the maximum frequency of the signal (Fmax). If the signal are not bandlimited, use a LPF (anti-aliasing filter). Typical Fs: 6 4 kHz (wireline phone apps) 16 kHz • Typical Fs: 6.4 kHz (wireline phone apps), 16 kHz (Hi-Fi); 44 kHz (music, CD quality) • For speech to be intelligible, you need at least an 8kHz SR. • The higher the SR the better the quality of the audio signal (usually). Concepts - Notations ! = 2 " f , in radians # = 2 " F analog frequency ( F is in Hz) ! = # T = # /Fs T is the sampling period and 5 Fs = 1/T = sampling frequency or sampling rate (SR) –I f SR = 50 kHz means 50 k samples/sec Useful Functions • Impulse Function : 10 n ± ² ´ [ n ] 1 6 () 00 n n ´ ± ³ µ n
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3 Useful Functions • Step Function : 10 () 00 n un n ± ² ³ ´ µ 7 • Properties: 0 ( ) k n k · ¸ ³ ³¹ º () ( 1 ) nu n ¹ Useful Functions • The Exponential Function in continuous time: 0 cos ( ) s in ( ) jt x tA e A tj A t » ³³ » ¼» 8 • In discrete time: If , is a decreasing sequence. n x nA ½ ³ ||1 µ x n Useful Functions • Sinusoidal Function 0 c o s ( ) x n n ¾¿ ³¼ À : frequency of the sinusoid, : phase o 9 I f || , || o j j eA A e ¾ ¿ ½½ 0 0 || || | || | [cos( ) sin( )] jn nj n n n AA e e Ae An j n ¿¾ ½¿ ¼ ³ ³ ¼ ¼
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4 Useful Functions • Sinusoidal Function
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113_1_113-midrev2010-toclass - Why Digital Signal...

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