Digital Signals - fac.ksu.edu.sa · Digital Signals 1 For floating point DS processor, the...
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Digital Signals
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For floating point DS processor, the amplitudes can be floating points.
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Common Digital Sequences
Unit‐impulse sequence:
Unit‐step sequence:
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Shifted Sequences
Shifted unit‐impulse Shifted unit‐step
Right shift by two samples
Left shift by two samples
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Sinusoidal and Exponential Sequences
Sinusoidal
Exponential
Exam
ple
Exam
ple
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Example 1
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Solution:
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Generation of Digital Signals
Let, sampling interval, Tt x(n): digital signalx(t): analog signal
Also
Example 2Convert analog signal x(t) into digital signal x(n), when sampling period is 125 microsecond, also plot sample values.
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Solution:
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Example 2 (contd.)
The first five sample values:
Plot of the digital sequence:
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Linear SystemSystem: A system that produces an output signal in response to an input signal.
Continuous system & discrete system.
Time, t
Sample number, n
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Linear Systems: Property 11. Homogeneity
2. Additivity
3. Shift invariance
Must for all linear systems
Must for DSP linear systems
Homogeneity: (deals with amplitude)
If x[n] y[n], then kx[n] ky[n]K is a constant
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Linear Systems: Property 2Additivity Homogeneity & Additivity
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Shift (time) Invariance
Linear Systems: Property 3
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Example 3Let a digital amplifier,
If the inputs are:
Outputs will be:
If we apply combined input to the system:
The output will be:
Individual outputs: )(40)(104)(4)(20)(102)(2
22
11
nnxnynunxny
+
X 10)(nx )(ny
X 10)()(1 nunx )(10)(1 nuny
X 10)()(2 nnx )(10)(2 nny
X 10)(2)(2 1 nunx )(20)(2 1 nuny
X 10)(4)(4 2 nnx )(40)(4 2 nny
Linear System
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Example 4
System)(nx )()( 2 nxny
System)()(1 nunx )()()( 2
1 nununy System
)()(2 nnx )()()( 22 nnny
If the input is:
Then the output is:
Individual outputs: )(2)(2)(2
)(4)(4)(42
22
112
nnxny
nunxny
+ Non Linear System
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Example 5 (a)Given the linear system , find whether the system is time invariant or not.
System)(1 nx )5(2)( 11 nxny
Let the shifted input be:
Therefore system output:
)5(2)( 11 nxnyShifting by n0 samples leads to
Equal
Time Invariant
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Solution:
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Example 5 (b)Given the linear system , find whether the system is time invariant or not.
System)(1 nx )3(2)( 11 nxny
Let the shifted input be:
Therefore system output:
)3(2)( 11 nxny Shifting by n0 samples leads to
NOT Equal
NOT Time Invariant
)3(2)( nxny
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Solution:
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Causality
Causal System:
Output y(n) at time n depends on current input x(n) at time n or previous inputs, such as x(n‐1), x(n‐2), etc.
Non Causal System:
Output y(n) at time n depends on future inputs, such as x(n+1), x(n+2), etc.
The non causal system cannot be realized in real time.
Example:
Example:
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Difference Equation
A causal, linear, and time invariant system can be represented by a difference equation as follows:
Outputs Inputs
After rearranging:
Finally:
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Example 6Identify non zero system coefficients of the following difference equations.
25.0 ,1 10 ab
5.0 ,1 10 bb
Solution:
Solution:
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System Representation Using Impulse Response
Impulse input with zero initial conditions
Impulse Response
Any input y(n) = x(n) h(n)
Convolution
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Example 7 (a)Given the linear time‐invariant system:
a.
Therefore,
b. c.
Solution:
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Example 7 (b)
a. Then
Infinite!
Solution:
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Example 7 (b) – contd.
b.
c.
Finite Impulse Response (FIR) system:
When the difference equation contains no previous outputs, i.e. ‘a’coefficients are zero. < See example 7 (a) >
Infinite Impulse Response (IIR) system:
When the difference equation contains previous outputs, i.e. ‘a’coefficients are not all zero. < See example 7 (b) >
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BIBO Stability
BIBO: Bounded In and Bounded Out
A stable system is one for which every bounded input produces a bounded output.
Let, in the worst case, every input value reaches to maximum value M.
Using absolute values of the impulse responses,
If the impulse responses are finite number, then output is also finite.
Stable system.
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BIBO Stability – contd.
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To determine whether a system is stable, we apply the following equation:
Impulse response is decreasing to zero.
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Example 8
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Given a linear system given by:
Which is described by the unit‐impulse response:
Determine whether the system is stable or not.
Solution:
Using definition of step function:
For a < 1, we know
Therefore The summation is finite, so the system is stable.
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Digital Convolution
The sequences are interchangeable.
Convolution sum requires h(n) to be reversed and shifted.
If h(n) is the given sequence, h(‐n) is the reversed sequence.
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][*][][*][ nxnhnhnx
Commutative
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Reversed Sequence
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Solution:a.
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Convolution Using Table MethodExample 9
Solution: Length = 3 Length = 3
Convolution length = 3 +3 – 1 = 5
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Convolution Using Table MethodExample 10
Solution:
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Convolution length = 3 + 2 – 1 = 4
Length = 3 Length = 2
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Associative: ])[*][(*][][*])[*][( ncnbnancnbna
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Convolution PropertiesCommutative: ][*][][*][ nanbnbna
Distributive: ])[][(*][][*][][*][ ncnbnancnanbna
Associative Distributive