113_1_Week02_print

113_1_Week02_print - 1 EE 113: Digital Signal Processing...

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1 EE 113: Digital Signal Processing Week 2 1. Discrete-time systems - definitions about systems and their characterizations 2. Impulse response sequence
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2 Discrete-time systems ± A system converts input to output: ± What makes a system special is that the input sequence x[n] must uniquely define the output signal y[n] x [ n ] DT System y [ n ] y [ n ] = S ( x [ n ])
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3 Discrete-time systems ± Is the following a system? y 2 [ n ] = |x [ n ]|
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4 Discrete-time systems Examples: y[n]=x[n] -> wire connection transmitting a signal from one point to another in a communication system y[n]=x[n-1] -> unit delay system y[n]=1/2(x[n-1]+x[n]) -> this system averages two successive samples of the input sequence
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5 Discrete-time systems ± For the previous systems, the output can be readily evaluated given the input sequence ± Assume x[n]=(-1) n -> see previous examples ± Sometimes, knowledge of the input sequence alone is not enough to evaluate the response of a system to it ± This is because the input-output transformation defines a class of systems rather than a single system Other examples of systems – Page 27, Chapter 4 (AHS) Example 2.11 and 2.12 (MB) Example on the board!
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6 Example: Accumulator ± Output accumulates all past inputs: x [ n ] y [ n ] + z -1 y [ n -1] y [ n ] = x [ A ] A =−∞ n = x [ A ] A n 1 + x [ n ] = y [ n 1] + x [ n ] Discussion
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7 Relaxed system ± A system that is initially at rest ± Defined as a system whose output sequence is 0 as long as the input is 0 Examples on the board!
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8 System property: Static or dynamic ± A system is static (= memoryless) if its output at time n depends only on its input at the same time instant n. ± Otherwise: dynamic (with memory) ± Examples: ± y[n]=ax[n], y[n]=cos[x[n]] ± y[n]=x[n 2 ] ± y[n]=x[n-1] ± {y[n]=y[n-1]+x[n], y[-1]=0}
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9 System property: Time (shift) invariance ± Time-shift of input causes same shift in output ± i.e. if x 1 [ n ] y 1 [ n ] then ± i.e. process doesn’t depend on absolute value of n x [ n ] = x 1 [ n n 0 ] y [ n ] = y 1 [ n n 0 ]
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10 Time-invariant example ± Hence ± If then ± Time (shift) invariant - parameters do not depend on n = [] 2 yn xn = 10 2 [ ] 2 [ ] y nn x x n n y nx n n =⋅ =−
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11 How do we prove that a system is time-variant? ± It is enough to find a counter-example!!
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12 Time-invariant counterexample ± Hence ± If then ± Not time (shift) invariant - parameters depend on n y [ n ] = n x [ n ] scaling by time index y 1 [ n n 0 ] = n n 0 ( ) x 1 [ n n 0 ] x [ n ] = x 1 [ n n 0 ] y [ n ] = n x 1 [ n n 0 ] Examples on the board!
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13 Another counterexample ± Upsampler: 10 [] [ ] x nx n n =− L ] [ n x y [ n ] y [ n ] = x [ n / L ], n = 0, ± L , ± 2 L , " otherwise 1 0 1 0 [ () / ] [ / / ] yn n x n n L xnL n L −= = Time variant!! 0
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This note was uploaded on 11/07/2009 for the course EE 113 taught by Professor Walker during the Spring '08 term at UCLA.

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113_1_Week02_print - 1 EE 113: Digital Signal Processing...

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