3/18/20161 Linear Time Invariant Systems Definitions A linear system may be defined as one which obeys the Principle of Superposition, which may be stated as follows: If an input consisting the sum of a number of signals is applied to a linear system, then the output is the sum, or superposition, of the system’s responses to each signal considered separately. A time-invariant system is one whose properties do not vary with time. The only effect of a time-shift on an input signal to the system is a corresponding time-shift in its output. A causal system is one if the output signal depends only on present and/or previous values of the input. In other words all real time systems must be causal; but if data were stored and subsequently processed at a later date, it need not be causal.

3/18/20162 The Unit Impulse Response The unit impulse is a single vertical line of zero width and a height of 1. This is also sometimes known as the Kronecker delta function  [n] is the symbol given to the line where  means infinitely small. n is the sampling period A shifted impulse such as  [n – 2] is the line shifted to the right 2 sampling periods. n  [n]  [n-2]

3/18/20163 Now consider the following signal: x[n] = 2  [n ] + 4  [n – 1] + 6  [n – 2] + 4  [n – 3] + 2  [n – 4] n  [n]  [n-1]  [n-2]  [n-3]  [n-4] X[n] Hence any sequence can be represented by the equation: = + x[-1]  [n + 1] + x[0]  [n] + x[1]  [n - 1] + x[2]  [n - 2] +……. x[k] is the height of each impulse, frequently known as the coefficient.  [n - k] is the time slot

3/18/20164 Impulse Response When the input to an FIR filter is a unit impulse sequence, x[n] =  [n], the output is known as the unit impulse response, which is normally donated as h[n]. A single impulse input yields the system’s impulse response

3/18/20165 A scaled impulse input yields a scaled response, due to the scaling property of the system's linearity.

3/18/20166 This now demonstrates the additivity portion of the linearity property of the system to complete the picture. Since any discrete-time signal is just a sum of scaled and shifted discrete-time impulses, we can find the output from knowing the input and the impulse response

3/18/20167 Convolution

3/18/20168 Convolution Convolution is a weighted moving average with one signal flipped back to front: The general expression for an FIR filter’s output is:- A tabulated version of convolution nn < n < 7 x[n] h[n]0321 h[0]x[n] h[1]x[n-1] h[2]x[n-2] h[3]x[n-3] y[n] h[0]x[n] = x[0] * h[0] + x[1] * h[0] + x[2] * h[0] + x[3] * h[0] + x[4] * h[0] h[0]x[n] =2 * * * * * 3 h[0]x[n] =

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10 FIR Filter Where Z -n is a delay of one sampling period a R is the coefficient (gain/attenuation of impulse  is the symbol for summation (adding) y(n)

3/18/ Now the following share prices were obtained from a weeks trading DayPeriodx(n)Price Monday0x(0)20 Tuesday1x(1)20 Wednesday2x(2)20 Thursday3x(3)12 Friday4x(4)40 Saturday5x(5)20 Sunday6x(6)20

3/18/ aRaR Value a0a a1a1 0.5 a2a x(0) = 20 x(-1)= 0 x(-2)= 0 Performing the multiplications and additions gives: y(0) = 0.25 x x x 0 = 5

3/18/ x(1) = 20 x(0)= 20 x(-1)= 0 It follows that: y(1) = 0.25 x x x 0 = 15 x(2) = 20 x(1)= 20 x(0)= 20 Giving: y(2) = 0.25 x x x 20 = 20 For Tuesday For Wednesday For Thursday x(3) = 12 x(2)= 20 x(1)= 20 Giving y(3) = 0.25 x x x 20 = 18

3/18/ Dayy(n) Monday5 Tuesday15 Wednesday20 Thursday18 Friday21 Saturday28 Sunday25

3/18/ The input impulse pulse train The impulse response of the filter h(n)

3/18/ y(5) = 0.25 x x x 12 = 28

3/18/ y(4) = 0.25 x x x 20 = 21 Dayy(n) Monday5 Tuesday15 Wednesday20 Thursday18 Friday21 Saturday28 Sunday25

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3/18/ Steps, Impulses and Ramps The unit step function u[n] is defined as: u[n] = 0, n < 0 u[n] = 1, n ≥ 0 This signal plays a valuable role in the analysis and testing of digital signals and processors. Another basic signal which is even more important than the unit step, is the unit impulse function d[n], and is defined as:  [n] = 0,n ≠ 0  [n] = 1,n = 0

3/18/ One further signal is the digital ramp which rises or falls linearly with the variable n. The unit ramp function r[n] is defined as: r[n] = n u[n]