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Unformatted text preview: Discretetime Systems The function of a discretetime system is to process a given input sequence to produce an output sequence. In practice, the input is a digital signal and the output is also a digital signal. Such a discretetime system is called as a digital filter. Example: Accumulator is the filter whose output is sum of all the input samples up to time instant n . ] [ ] [ ] [ ] [ 1 n x k x k x n y n k n k + = = ∑ ∑∞ =∞ = ] [ ] 1 [ n x n y + = The inputoutput equation can be expressed as , ] [ ] 1 [ ] [ ≥ + = ∑ = n k x y n y n k This expression is used for a causal input sequence with initial condition y [ 1]. Example:Movingaverage filter (runningaverage filter) is a simple but useful filter for smoothing the input signal. The output of the running average filter can be written as ∑ = + = 1 ] [ 1 ] [ K k n k n x K n y , n = offset e.g. K =3, n =0 ( 29 ] 2 [ ] 1 [ ] [ 3 1 ] [ + + = n x n x n x n y n321 1 2 3 4 5 6 7 x[n] 2 4 6 4 2 y[n] 2/3 2 4 14/3 4 2 2/3 Further illustration of FIR filtering Consider the signal ] [ n x given by otherwise n n n x n 40 ) 4 / 8 / 2 cos( 5 . ) 02 . 1 ( ] [ ≤ ≤ + + = π π The first term of the signal is the component of interest while the second term is the noise or interference. Now we use causal FIR filter of M = 2, 6 to process the signal in order to observe the smoothing effect of using different FIR filters. Input signal: 40 ) 4 / 8 / 2 cos( 5 . ) 02 . 1 ( ] [ ≤ ≤ + + = n for n n x n π π 1 0 2 0 3 0 4 0 5 0 0 . 5 1 1 . 5 2 2 . 5 3 n x[n] Output of 3point runningaverage filter 1 0 2 0 3 0 4 0 5 0 0 . 5 1 1 . 5 2 2 . 5 3 n output Output of 7point runningaverage filter 1 0 1 0 2 0 3 0 4 0 5 0 0 . 5 1 1 . 5 2 2 . 5 n output Observations: (A) 3point runningaverage filter (1) The input sequence x[n] is zero prior to n=0, and it is clear that the output of the runningaverage filter is zero for n<0. (2) The output becomes nonzero at n=0 and the 3point averager “runs onto” the input sequence during the interval 2 ≤ ≤ n . For 40 2 ≤ ≤ n , the input samples “fill up” the averager. (3) The averager “runs off” the input sequence at the end. (4) It is observed that the size of the sinusoidal component is reduced. (B) 7point runningaverage filter Similar results are observed. The “runonto” and “runoff” regions are getting longer and the sinusoidal component is greatly reduced. (C) Repeat the filtering using a 3tap filter: ] 2 [ ] 1 [ )) 4 / cos( 2 ( ] [ ] [ + ⋅ π = n x n x n x n y 1 0 2 0 3 0 4 0 5 0 1 0 . 5 0 . 5 1 1 . 5 2 2 . 5 3 i n d e x n output Comment the difference between the two filters. Classification of Discretetime Systems (i) Linear system A system is said to be linear if y 1 [ n ] and y 2 [ n ] are the responses (outputs) to the input sequences x 1 [ n ] and x 2 [ n ], respectively, then for an input ] [ ] [ ] [ 2 1 n x n x n x β α + = the response is given by ] [ ] [ ] [ 2 1 n y n y n y β α + = This is the superposition property that holds for any arbitrary constants...
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 Fall '10
 ShuHungLeung
 Digital Signal Processing, Signal Processing, LTI system theory, Impulse response, Finite impulse response, discretetime systems

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