DSP_Chap2_Part1.pdf - 2.0 Introduction Chapter 2...

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12/2/2018 1 Chapter 2 Discrete-Time Signals and Systems ECE107 1 2.0 Introduction Signal : something conveys information, represented mathematically as functions of one or more independent variables. Classified as: Continuous-time (analog) signals , discrete-time signals , digital signals Signal-processing systems are classified along the same lines as signals: Continuous-time (analog) systems , discrete-time systems , digital systems ECE107 2 2.1 Discrete-Time Signals: Sequences Discrete-Time signals are represented as In sampling of an analog signal x a ( t ): 1/T (reciprocal of T) : sampling frequency   integer : , , n n n x x   period sampling T nT x n x a : , Cumbersome, so just use   x n ECE107 3 Figure 2.1 Graphical representation of a discrete-time signal Abscissa: continuous line : is defined only at discrete instants   x n ECE107 4 Figure 2.2 EXAMPLE Sampling the analog waveform ) ( | ) ( ] [ nT x t x n x a nT t a ECE107 5 Sum of two sequences Product of two sequences Multiplication of a sequence by a number α Delay (shift) of a sequence Basic Sequence Operations ] [ ] [ n y n x integer : ] [ ] [ 0 0 n n n x n y ] [ ] [ n y n x ] [ n x ECE107 6
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12/2/2018 2 Basic sequences Unit sample sequence (discrete-time impulse, impulse, Unit impulse )   0 1 0 0 n n n ECE107 7 Basic sequences [ ] [ ] [ ] k x n x k n k  arbitrary sequence   7 2 1 3 7 2 1 3 n a n a n a n a n p A sum of scaled, delayed impulses ECE107 8 Basic sequences Unit step sequence 0 0 0 1 ] [ n n n u   [ ] n k k u n  0 [ ] [ 1] [ 2] ] [ [ ] k n n n u n n k ] 1 [ ] [ ] [ n u n u n First backward difference     0, 0 , 1, 0 0 0 1 0 since n k when n k when n k k k  ECE107 9 Basic Sequences Exponential sequences n A n x ] [ A and α are real: x[n] is real A is positive and 0< α <1, x[n] is positive and decrease with increasing n -1< α <0, x[n] alternate in sign, but decrease in magnitude with increasing n : x[n] grows in magnitude as n increases 1 ECE107 10 EX. 2.1 Combining Basic sequences 0 0 0 ] [ n n A n x n If we want an exponential sequences that is zero for n <0, then ] [ ] [ n u A n x n Cumbersome simpler ECE107 11 Basic sequences Sinusoidal sequence n all for n w A n x 0 cos ] [ ECE107 12
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12/2/2018 3 Exponential Sequences 0 jw e j e A A n w A j n w A e A e e A A n x n n n w j n n jw n j n 0 0 sin cos ] [ 0 0 1 1 1 Complex Exponential Sequences Exponentially weighted sinusoids Exponentially growing envelope Exponentially decreasing envelope 0 [ ] jw n x n Ae is refered to ECE107 13
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  • Summer '15
  • LTI system theory, N0, Continuous-time

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