Discrete - Time signals: sequences Discreet -Time signals are represented mathematically as sequences of numbers The sequence is denoted [], and it is written formally as = ; −∞ < < ∞ where n is an integer number In practice sequences arises from the periodic sampling of an analog signal 1
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Discrete-Time signals:
sequencesDiscreet-Time signals are represented
mathematically as sequences of numbers
The sequence is denoted 𝑥[𝑛], and it is
written formally as
𝑥 = 𝑥 𝑛 ; −∞ < 𝑛 < ∞
where n is an integer number
In practice sequences arises from the
periodic sampling of an analog signal
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Discrete-Time signals:
sequencesIn this case the numeric value of the nth
number in the sequence is equal to the
value of the analog signal, 𝑥𝑎(𝑡), at time
𝑛𝑇𝑥 𝑛 = 𝑥𝑎[𝑛𝑇]
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Examples of sequences
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Basic sequences and sequence
operationThe product and sum of two sequences x[n]
and 𝑦[𝑛] are defined as the sample by
sample product and sum
Multiplication of a sequence 𝑥[𝑛] by a
number 𝛼 is defined as the multiplication of
each sample value by 𝛼
A sample 𝑦[𝑛] is said to be delayed or shifted
version of 𝑥[𝑛] if 𝑦 𝑛 = 𝑥[𝑛 −
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MATLAB exercise
Record a voice signal using the
audiorecorder function for 5 seconds with
the following specifications
sampling frequency of 44100
Number of quantization bits 16
Number of channels = 1 for mono
Try to multiply the recorded samples by a
scaling factor of 𝛼 = 0.1 then by 𝛼 = 2 Play
the signal and hear the voice
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Special sequences Unit sample
sequenceUnit sample sequence is defined as the
sequence
One of the important aspects of the impulse
sequence is that an arbitrary sequence can
be presented as a sum of scaled, delayed
impulses as shown in the next slide
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Special sequences Unit sample
sequence
In general any sequence can be written as
𝑥 𝑛 = 𝑘=−∞∞ 𝑥 𝑘 𝛿[𝑛 − 𝑘]
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Special sequences Unit step
sequenceThe unit step sequence is given by
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Special sequences Unit step
sequenceThe unit step sequence is given by
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Special sequences Unit step
sequenceThe unit step sequence in terms of
delayed impulses can be written as 𝑢 𝑛 =𝛿 𝑛 + 𝛿 𝑛 − 1 + 𝛿 𝑛 − 2 +⋯ = 𝑘=0∞ 𝛿 𝑛 − 𝑘
Note that the impulse sequence can be
expressed as the first backward difference
of the unit step sequence
𝛿 𝑛 = 𝑢 𝑛 − 𝑢[𝑛 − 1]
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Special sequences exponential
sequencesExponential sequence are important in
representing and analyzing linear time
invariant systems
The general form of an exponential sequence
is given by 𝑥 𝑛 = 𝐴𝛼𝑛
If 𝐴 and 𝛼 are real then the sequence is real
If 0 < 𝛼 < 1 and 𝐴 is positive then the
sequence values are positive and decreasing
with increasing 𝑛
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Special sequences exponential
sequencesGraphical representation of exponential
sequence
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Special sequences sinusoidal
sequencesThe general form of sinusoidal sequence is
given by 𝑥 𝑛 = 𝐴𝑐𝑜𝑠(𝜔0𝑛 + ∅) as shown
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Special sequences sinusoidal and
complex exponential sequence
The exponential sequence 𝑥 𝑛 = 𝐴𝛼𝑛 with
complex 𝛼 has a real and imaginary parts that
are exponentially weighted sinusoids
If 𝛼 = 𝛼 𝑒𝑗𝜔0 and 𝐴 = 𝐴 𝑒𝑗∅ then the sequence
can be expressed in either one of the following
forms
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Notes about sequences
When discussing either complex exponential signals of the form 𝑥 𝑛 = 𝐴𝑒𝑗𝜔0𝑛 or real sinusoidal signal of the form 𝑥 𝑛 = 𝐴𝑐𝑜𝑠 𝜔0𝑛 + ∅ we need only to consider frequencies in an interval of length of 2𝜋only because
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Periodic sequence
A periodic sequence is a sequence that
satisfies the following equation
𝑥 𝑛 = 𝑥[𝑛 + 𝑁],
Where 𝑁 is an integer number
If this condition is tested for the discrete
time sinusoids, then
Which requires
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Periodic sequence
Where 𝑘 is an integer
A similar statement holds for the complex
exponential
Where 𝑁 is an integer number
Again
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Example
Determine if the following sequences are
periodic or not. If the sequence is periodic
find its period
a) 𝑥1 𝑛 = cos𝑛𝜋
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b) 𝑥2 𝑛 = cos3𝑛𝜋
4
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solution
a) For the first sequence we have 𝜔0𝑁 =
2𝜋k or 𝜋
4𝑁 = 2𝜋𝑘 → 𝑁 = 8𝑘 since 𝑁 is an
integer value the sequence is periodic
b) For the second sequence 𝜔0𝑁 = 2𝜋𝑘 or 3𝜋
4𝑁 = 2𝜋𝑘 → 𝑁 =
8
3𝑘 since 𝑁 is not an
integer value for 𝑘 = 1 the sequence is
aperiodic if 𝑁 = 8
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2.2 Discrete time systems
A discrete-time system is a system that
maps an input sequence with an output
sequence 𝑦 𝑛 = 𝑇{𝑥 𝑛 }
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Discrete time system examples
There are many systems will be
investigated through out this course
Examples of these systems are
1. The ideal delay system which is described
mathematically by 𝑦 𝑛 = 𝑥 𝑛 − 𝑛𝑑 , −∞ <𝑛 < ∞
2. Moving average system which is described
mathematically by 1
𝑀1+𝑀2+1 𝑘=−𝑀1𝑀2 𝑥[𝑛 − 𝑘]
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Discrete time system
classificationsSystems can be classifieds into one of the
following categories
1. Memoryless Systems. A system is classified
into memoryless system if the output 𝑦 𝑛 at
every value of 𝑛 depends only on the input
of 𝑥[𝑛] at the same value of 𝑛. An example
of a memoryless system is the squarer
system described by 𝑦 𝑛 = 𝑥[𝑛] 2
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Discrete time system
classifications2. Linear systems. Any system satisfies the
superposition and the scaling property is
classifieds as a linear system. As an
example of a linear system is the
accumulator system described by
𝑦 𝑛 = 𝑘=−∞𝑛 𝑥[𝑘]
3. Time-invariant system is a system for which
a time shift or delay of the input sequence
causes a corresponding shift in the output
sequence
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Discrete time system
classificationsExample show that the accumulator system
𝑦 𝑛 = 𝑘=−∞𝑛 𝑥[𝑘] is a time invariant system
solution
Assume that the input to the accumulator is
𝑥1 𝑛 = 𝑥[𝑛 − 𝑛0], then its output is 𝑦1 𝑛 = 𝑘=−∞𝑛 𝑥1[𝑘] = 𝑘=−∞
𝑛 𝑥[𝑘 − 𝑛0]
Let 𝑘1 = 𝑘 − 𝑛0This means that
𝑦1 𝑛 = 𝑘=−∞𝑛−𝑛0 𝑥[𝑘1] = y[n − 𝑛0]
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Discrete time system
classifications4. Causality, a system is causal if the output
sequence value at the index 𝑛 − 𝑛0 depends
only on the input sequence values for 𝑛 ≤ 𝑛0For example the forward difference system
described by 𝑦 𝑛 = 𝑥 𝑛 + 1 − 𝑥 𝑛 is not causal
because the current value of the output depends on
future value of the input
Another example is the backward difference system
𝑦 𝑛 = 𝑥 𝑛 − 𝑥[𝑛 − 1] is a causal system since the
output depends only on the present and past
values of the input
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Discrete time system
classifications5. Stability, a system is stable if and only if
every bounded input sequence produces a
bounded output sequence
Such a system is called BIBO
in equation form
𝑥 𝑛 ≤ 𝐵𝑥 < ∞ → 𝑦 𝑛 ≤ 𝐵𝑦 < ∞
In general any sequence that has the form
𝑦 𝑛 = 𝑘=−∞𝑛 𝑥[𝑘] < ∞ is stable system
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Linear time-invariant system
The linear time-invariant system is an
important system since many of the system
we deal with in signal processing are of this
type
The output sequence in response to the
input sequence applied to the input of the
linear time-invariant system is given by the
convolutional sum 𝑦 𝑛 = 𝑘=−∞∞ 𝑥 𝑘 ℎ[𝑛 − 𝑘]
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Linear time-invariant system
In order to compute the convolution we
draw both ℎ 𝑛 − 𝑘 and 𝑥[𝑘] sequences as
shown below
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Linear time-invariant system
From the Figure, we have 𝑦 𝑛 = 0 𝑓𝑜𝑟 𝑛 <0
The next sequence interval is shown by the
next graph that is 0 ≤ 𝑛 ≤ 𝑁 − 1
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Linear time-invariant system
The output sequence for this interval is
given by
This equation can be solved analytically by
using the geometric series expansion
𝑘=𝑁1
𝑁2
𝑎𝑘 =𝑎𝑁1 − 𝑎𝑁2 +1
1 − 𝑎
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Linear time-invariant system
The output sequence for this interval is
given by
This equation can be solved analytically by
using the geometric series expansion
𝑘=𝑁1
𝑁2
𝑎𝑘 =𝑎𝑁1 − 𝑎𝑁2 +1
1 − 𝑎
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Convolution example
Which yields the following result
𝑦 𝑛 =
𝑘=0
𝑛
𝑎𝑘 =1 − 𝑎𝑛+1
1 − 𝑎𝑓𝑜𝑟 0 ≤ 𝑛 ≤ 𝑁 − 1
We consider the next interval when 0 < 𝑛 −𝑁 + 1
The output sequence is given by
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Convolution example
Which yields the following result
The final answer for the output sequence for
these three intervals is given by
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Convolution example
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Convolution in Matlab
Convolution can be accomplished easily in
matlab by using the function conv(u,v)
The above example can be solved easily in
matalb by using the following code in matlab
n=1:10;
h=ones(1,5);
x=0.4.^n;
Y=conv(x,h);
stem(y);
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2.4 Properties of linear time
invariant systemThe output sequence 𝑦[𝑛] of all LTI are