1. 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.

2. 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]

3. 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

4. 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

5. of the system's linearity.
A scaled impulse input yields a scaled response, due to the scaling property
of the system's linearity.
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6. 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

7. Convolution

8. 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
h[0]x[n] =

10. 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)

11. 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)

12. 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 x x 0 = 5

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

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

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

16. y(5) = 0.25 x x x 12 = 28

17. y(4) = 0.25 x 40 + 0.5 x 12 + 0.25 x 20 = 21 Day y(n) Monday 5 Tuesday15
Wednesday 20
Thursday 18
Friday 21
Saturday 28
Sunday 25

18. 7/3/2018

19. End

20. 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
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21. 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]
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