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.

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 d[n] is the symbol given to the line where d means infinitely small. n is the sampling period A shifted impulse such as d[n – 2] is the line shifted to the right 2 sampling periods. n -2 -1 1 2 3 4 5 6 d[n] d[n-2]

Now consider the following signal: x[n] = 2d[n ] + 4d[n – 1] + 6d[n – 2] + 4d[n – 3] + 2d[n – 4] n -2 -1 1 2 3 4 5 6 2d[n] 4d[n-1] 6d[n-2] 4d[n-3] 2d[n-4] X[n] Hence any sequence can be represented by the equation: = + x[-1]d[n + 1] + x[0]d[n] + x[1]d[n - 1] + x[2]d[n - 2] +……. x[k] is the height of each impulse, frequently known as the coefficient. d [n - k] is the time slot

Impulse Response When the input to an FIR filter is a unit impulse sequence, x[n] = d[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

of the system's linearity. A scaled impulse input yields a scaled response, due to the scaling property of the system's linearity. 7/3/2018

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

Convolution

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 n n < 0 1 2 3 4 5 6 7 n < 7 x[n] h[n] -1 h[0]x[n] 12 18 h[1]x[n-1] -2 -4 -6 h[2]x[n-2] 8 h[3]x[n-3] y[n] 10 16 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 + 4 * 3 + 6 * 3 + 4 * 3 + 2 * 3 h[0]x[n] = 6 + 12 + 18 + 12 + 6

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

Now the following share prices were obtained from a weeks trading Day Period x(n) Price Monday x(0) 20 Tuesday 1 x(1) Wednesday 2 x(2) Thursday 3 x(3) 12 Friday 4 x(4) 40 Saturday 5 x(5) Sunday 6 x(6)

Performing the multiplications and additions gives: aR Value a0 0.25 a1 0.5 a2 x(0) = 20 x(-1) = 0 x(-2) = 0 Performing the multiplications and additions gives: y(0) = 0.25 x 20 + 0.5 x 0 + 0.25 x 0 = 5

For Tuesday x(1) = 20 x(0) = 20 x(-1) = 0 It follows that: y(1) = 0.25 x 20 + 0.5 x 20 + 0.25 x 0 = 15 For Wednesday x(2) = 20 x(1) = 20 x(0) = 20 Giving: y(2) = 0.25 x 20 + 0.5 x 20 + 0.25 x 20 = 20 For Thursday x(3) = 12 x(2) = 20 x(1) = 20 Giving y(3) = 0.25 x 12 + 0.5 x 20 + 0.25 x 20 = 18

Day y(n) Monday 5 Tuesday 15 Wednesday 20 Thursday 18 Friday 21 Saturday 28 Sunday 25

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

y(5) = 0.25 x 20 + 0.5 x 40 + 0.25 x 12 = 28

y(4) = 0.25 x 40 + 0.5 x 12 + 0.25 x 20 = 21 Day y(n) Monday 5 Tuesday 15 Wednesday 20 Thursday 18 Friday 21 Saturday 28 Sunday 25

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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: d[n] = 0, n ≠ 0 d[n] = 1, n = 0 7/3/2018

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] 7/3/2018