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

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Discrete-Time signals: sequences

In this case the numeric value of the nth number in the sequence is equal to the value of the analog signal, , at time

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Examples of sequences

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Basic sequences and sequence operation

The product and sum of two sequences x[n] and are defined as the sample by sample product and sum

Multiplication of a sequence by a number is defined as the multiplication of each sample value by

A sample is said to be delayed or shifted version of if

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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 then by Play the signal and hear the voice

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Special sequences Unit sample sequence

Unit 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

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Special sequences Unit step sequence

The unit step sequence is given by

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Special sequences Unit step sequence

The unit step sequence is given by

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Special sequences Unit step sequence

The unit step sequence in terms of delayed impulses can be written as

Note that the impulse sequence can be expressed as the first backward difference of the unit step sequence

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Special sequences exponential sequences

Exponential sequence are important in representing and analyzing linear time invariant systems

The general form of an exponential sequence is given by

If and are real then the sequence is real

If and is positive then the sequence values are positive and decreasing with increasing

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Special sequences exponential sequences

Graphical representation of exponential sequence

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Special sequences sinusoidal sequences

The general form of sinusoidal sequence is given by as shown

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Special sequences sinusoidal and complex exponential sequence

The exponential sequence with complex has a real and imaginary parts that are exponentially weighted sinusoids

If and 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 or real sinusoidal signal of the form we need only to consider frequencies in an interval of length of only because

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Periodic sequence

A periodic sequence is a sequence that satisfies the following equation

,

Where is an integer number

If this condition is tested for the discrete time sinusoids, then

Which requires

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Periodic sequence

Where is an integer

A similar statement holds for the complex exponential

Where 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

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solution

a) For the first sequence we have or since is an integer value the sequence is periodic

b) For the second sequence or since is not an integer value for the sequence is aperiodic if

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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

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Discrete time system examples

There are many systems will be investigated through out this course

Examples of these systems are1. The ideal delay system which is described

mathematically by

2. Moving average system which is described mathematically by

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Discrete time system classifications

Systems can be classifieds into one of the following categories

1. Memoryless Systems. A system is classified into memoryless system if the output at every value of depends only on the input of at the same value of . An example of a memoryless system is the squarer system described by

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Discrete time system classifications

2. 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

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 classifications

Example show that the accumulator system is a time invariant system

solution

Assume that the input to the accumulator is , then its output is

Let

This means that

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Discrete time system classifications

4. Causality, a system is causal if the output sequence value at the index depends only on the input sequence values for

For example the forward difference system described by is not causal because the current value of the output depends on future value of the input

Another example is the backward difference system 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 classifications

5. 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

In general any sequence that has the form 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

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Linear time-invariant system

In order to compute the convolution we draw both and sequences as shown below

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Linear time-invariant system

From the Figure, we have

The next sequence interval is shown by the next graph that is

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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

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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

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Convolution example

Which yields the following result

We consider the next interval when

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 matlabn=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 system

The output sequence of all LTI are described by the convolution sum

Where is the impulse response of the LTI system

This means that is a complete characterization of the properties of a specific LTI system

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Properties of the convolution sum

commutative

Distribution over addition

Associative

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Graphical representation of combined LTI systems

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Cascaded systems can be presented by a single system whose impulse response is given by . Cascaded systems satisfy the convolution commutative property

Systems connected in parallel can be replaced by a single system whose .

Stability and causality in terms of

LTI are stable if and only if there impulse response is absolutely summable i.e.

LTI is causal if

Causality means that the difference equations describing the system can be solved recursively

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FIR systems – reflected in the h[n]

Ideal delay

Forward difference

Backward difference

Finite-duration impulse response (FIR) system are characterized by an impulse response has that has only a finite number of nonzero samples

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IIR systems – reflected in the

Accumulator

Infinite duration impulse response (IIR) system has whose duration extends to infinity

Stability

FIR systems always are stable, if each value of values is finite in magnitude

IIR systems can be stable, e.g.

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Cascading system examples

Determine if the following system is causal or not

Solution Since the impulse response of the cascaded

system satisfy the resulting cascaded system is stable

Any FIR system can be made causal by cascading it with a sufficiently long delay

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Cascading system examples

Determine the impulse response of the following cascaded systems

An inverse system is given by

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Linear constant-coefficient difference equations

The Nth order linear constant coefficient equations are a subclass of linear time invariant systems

The general form of these equations is

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Example of difference equations

Write the accumulator system in terms of difference equations

Solution The accumulator equation is given by

The output for can be written as

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Example of difference equations

Now the output sequence can be written as

Or alternatively it can be written as

If we compare the last equation with we find that

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Example of difference equations

The difference equations gives a better understanding of how we can be implement the accumulator system in this example

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Solving the Linear constant coefficient difference equationsDifference equations are similar to differential equations in continuous systems

The solution for the difference equations is composed from the homogeneous and particular solutions as described mathematically by

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Solving the Linear constant coefficient difference equationsThe homogeneous solution is obtained with

This means that the difference equation reduces to

Since has undetermined coefficients, a set of auxiliary conditions is required for the unique specification of for a given

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Solving the Linear constant coefficient difference equationsThese auxiliary conditions might consist of specifying fixed values of at specific values of , such as

The above step results in a set of linear equations for the undetermined coefficients, which can be solved to produce the required coefficients

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Recursive solution of the difference equations

The output samples for can be computed recursively by rearranging the difference equation as shown below

If the input , together with a set of auxiliary values is specified then the output can be computed

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Recursive solution of the difference equations

With available can be computed

To generate values of for , we can rearrange the linear constant coefficient difference equation as shown below

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Recursive computation example

Example: solve the following difference equation recursively

Assume that the input is and

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Recursive computation example

When , we can use recursive computation as follows

Let then

Since , then

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Recursive computation example

Next we do the same procedure when

To determine the output for , we express the difference equations in the form

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Recursive computation example

If we use the auxiliary conditions , we can compute for as follows

By combining the solutions for and , we got the following solution

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2.6 Frequency-domain representation of discrete time

signals and systemsThe frequency response of a given system with impulse response of is defined by

The output of any system characterized by its frequency response is given by

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Frequency response of the ideal delay system

Example determine the frequency response of an ideal delay system described by the following equation

Solution

To find the frequency response we first find the impulse response of the system which can be found by substituting

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Frequency response of the ideal delay system

This means that

Now the frequency response is given by

can be written in rectangular form as illustrated below

, from Euler identity

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2.7 Representation of sequences by Fourier transforms

In order to represent a given sequence by its Fourier transform we can use the following equation

However the inverse Fourier transform is given by

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Representation of sequences by Fourier transforms

For the discrete time signals, the value of is restricted to an interval of

The low frequency component of discrete time signals are located around

The high frequency component are located around

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Convergence of the Fourier transform

In general not all the signals have Fourier transform

Only the absolutely summable signals have their Fourier transform exits

Absolutely summable signals are signals satisfying the following condition

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Example

Determine if has a Fourier transform or not. If the Fourier transform exist, find the value of

Solution

The summation

If and only if this means that the discrete Fourier transform exists only for

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Example

The summation

If and only if this means that the discrete Fourier transform exists only for

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