Bài giảng xử lý tín hiệu số signal and system in time domain ngô quốc cường

2.1.2 Classification of discrete time signal • Energy signals and power signal – The energy E of a signal xn is given: – If E is finite, xn is call an energy signal... 2.2 Discrete time

Xử lý tín hiệu số Signal and System in Time Domain

Ngô Quốc Cường

Ngô Quốc Cường

ngoquoccuong175@gmail.com

sites.google.com/a/hcmute.edu.vn/ngoquoccuong

Signal and System in Time Domain

• Discrete time signals

• Discrete time systems

• LTI systems

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2.1 Discrete - time signals

• A discrete time signal x(n) is a function of an independent

variable that is integer

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2.1 Discrete - time signals

• Alternative representation of discrete time signal:

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2.1 Discrete - time signals

• Alternative representation of discrete time signal:

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2.1.1 Some elementary signals

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2.1.1 Some elementary signals

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2.1.1 Some elementary signals

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2.1.1 Some elementary signals

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2.1.2 Classification of discrete time signal

• Energy signals and power signal

– The energy E of a signal x(n) is given:

– If E is finite, x(n) is call an energy signal

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2.1.2 Classification of discrete time signal

• Energy signals and power signal

– The average power P of a signal x(n) is defined:

– If P is finite (and nonzero), x(n) is called a power signal

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2.1.2 Classification of discrete time signal

• Energy signals and power signal

– Example: the average power of the unit step signal is:

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2.1.2 Classification of discrete time signal

• Periodic signals and aperiodic signals

– A signal x(n) is periodic with period N (N >0) if and only if

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2.1.2 Classification of discrete time signal

• Symmetric (even) and antisymmetric (odd) signals

– A real value signal x(n) is call symmetric if

– A signal is call antisymmetric if

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2.1.2 Classification of discrete time signal

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• Symmetric (even) and antisymmetric (odd) signals

2.1.2 Classification of discrete time signal

– The even signal component is formed by adding x(n) to n) and dividing by 2

x(-– Odd signal component

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2.1.3 Simple manipulations of signals

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• Transformation of time

– A signal x(n) may be shifted by replacing n bay n-k

• k is positive number: delay

• k is negative number: advance – Folding: replace n by -n

– Time scaling: replace n by cn (c is an integer)

2.1.3 Simple manipulations of signals

• Transformation of time

– Find x(n-3) and x(n+2) of x(n)

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2.1.3 Simple manipulations of signals

• Transformation of time

– Find x(-n) and x(-n+2) of x(n)

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2.1.3 Simple manipulations of signals

• Transformation of time

– Show the graphical representation of y(n) = x(2n), where x(n) is

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2.1.3 Simple manipulations of signals

• Addition, multiplication, and scaling

– Amplitude scaling

– Sum

– Product

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Exercises

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Exercises

• x(n) is illustrated in the figure

• Sketch the following signals

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2.2 Discrete time systems

• Device or algorithm that performed some prescribed operation on discrete time signal

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2.2 Discrete time systems

• Determine the response of the following systems to the input signal

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2.2 Discrete time systems

• Block diagram representation

– An adder

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2.2 Discrete time systems

• Block diagram representation

– A constant multiplier

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2.2 Discrete time systems

• Block diagram representation

– A signal multiplier

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2.2 Discrete time systems

• Block diagram representation

– A unit delay element

– A unit advance element

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2.2 Discrete time systems

• Block diagram representation

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2.2 Discrete time systems

• Classification of discrete time systems

– Static versus dynamic systems

– Time invariant versus time variant systems

– Linear versus nonlinear systems

– Causal versus noncausal systems

– Stable versus unstable systems

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2.2 Discrete time systems

• Classification of discrete time systems

– Static versus dynamic systems

• Static: output at any instant n depends at most on the input sample at the same time – memoryless

• Dynamic: to have memory

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2.2 Discrete time systems

• Classification of discrete time systems

– Time invariant versus time variant systems

• Time invariant: input – output characteristics do not change with time

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2.2 Discrete time systems

• Classification of discrete time systems

– Linear versus nonlinear systems

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2.2 Discrete time systems

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2.2 Discrete time systems

• Classification of discrete time systems

– Causal versus noncausal systems

• The output of the system at any time n depends only on present and past inputs but does not depend on future inputs

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2.2 Discrete time systems

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2.2 Discrete time systems

• Classification of discrete time systems

– Stable versus unstable systems

• An arbitrary relaxed system is said to be bounded input bounded output stable if and only if every bounded input produces a bounded output

𝑥 𝑛 ≤ 𝑀𝑥 < ∞

𝑦 𝑛 ≤ 𝑀𝑦 < ∞

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2.2 Discrete time systems

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2.2 Discrete time systems

• Interconnection of discrete time systems

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2.3 Analysis of discrete time LTI systems

• LTI: Linear Time Invariant

• 2 methods:

– Solve the difference equation

– Decompose the input signal into a sum of elementary signals Using the linearity property, the responses of the system to the elementary signals are added to obtain the total response of the system

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2.3 Analysis of discrete time LTI systems

• Resolution of discrete time signal into impulses

• Example:

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2.3 Analysis of discrete time LTI systems

• Response of LTI system to arbitrary input

– Denote the response y(n, k) of the system to unit sample sequence at n = k by symbol h(n, k)

– The response of the system to x(n)

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2.3 Analysis of discrete time LTI systems

• Response of LTI system to arbitrary input: convolution

– The formula reduces to

– The response at n = n0 is given as

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2.3 Analysis of discrete time LTI systems

• Response of LTI system to arbitrary input

– Summarize

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2.3 Analysis of discrete time LTI systems

• Response of LTI system to arbitrary input

– Example

– The output is

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Exercise

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Exercise

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Exercise

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2.3 Analysis of discrete time LTI systems

• Properties of convolution and interconnection

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2.3 Analysis of discrete time LTI systems

• Causal linear time invariant system

• An LTI system is causal if and only if its impulse response is zero for n<0

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2.3 Analysis of discrete time LTI systems

• Stability of linear time invariant system

– A linear time invariant system is stable if its impulse

response is summable

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• LTI system can be characterized in terms of its impulse response h(n)

– Finite duration impulse response

– Infinite duration impulse response

• Causal FIR system

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• Practical DSP methods fall in two basic classes:

– Block processing methods

– Sample processing methods

• In block processing methods, the data are collected and processed in blocks

• In sample processing methods, the data are processed one

at a time Sample processing methods are used primarily in real-time applications

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Block processing

• In many practical applications, we sample our analog input signal (in accordance with the sampling theorem requirements) and collect a finite set of samples, say L samples, representing a finite time record of the input signal The duration of the data record in seconds will be:

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Block processing

• The direct and LTI forms of convolution given by

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Block processing

• Direct Form

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Block processing

• Direct Form

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Block processing

• Convolution table

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Block processing

• Convolution table

– Folding the table,

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Block processing

• LTI form

– The effect of the filter is to replace each delayed impulse

by the corresponding delayed impulse response

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Block processing

• LTI form

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Block processing

• Matrix form

– The convolutional equations can also be written in the

linear matrix form

– The filter matrix H must be rectangular with dimensions

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Block processing

• Matrix form

– There is also an alternative matrix form written as follows:

– the data matrix X has dimension:

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Block processing

• Flip and Slide form

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Block processing

• Overlap-Add block

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Block processing

• Overlap-Add block

– aligning the output blocks according to their absolute

timings and adding them up

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Problems

• Compute the convolution, y = h ∗ x, of the filter and input

• Using the following three methods:

– (a) The convolution table

– (b) The LTI form of convolution, arranging the

computations in a table form

– (c) The overlap-add method of block convolution with

length-3 input blocks Repeat using length-5 input blocks

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2.4 Discrete time systems described by

difference equations

• The practical implementation of the IIR system is impossible since it requires an infinite number of memory locations, multiplications, and additions

• Practical and computationally efficient means: difference equations

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2.4 Discrete time systems described by

difference equations

• Recursive and non-recursive system

– Compute the cumulative average of a signal x(n) defined

2.4 Discrete time systems described by

difference equations

• Recursive and non-recursive system

– Non-recursive system: depends only on the present and the past inputs

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2.4 Discrete time systems described by

difference equations

• Recursive and non-recursive system

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2.4 Discrete time systems described by

difference equations

• The general form:

• N: the order of the difference equation = the order of the system

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2.4 Discrete time systems described by

difference equations

• Solution of linear constant coefficient difference equation

– Direct method

– Indirect method (z - transform)

• The direct solution method assumes that the total solution is the sum of two parts:

– y h (n): homogeneous solution

– y p (n): particular solution

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2.5 Structure for the realization of LTI systems

• Consider the 1st order system

• This realization uses separate delays for both input and output, called Direct Form 1 structure

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2.5 Structure for the realization of LTI systems

• Interchange the order of the recursive and non-recursive

systems

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2.5 Structure for the realization of LTI systems

• Two delay elements can be merged into one delay

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Direct Form 2 structure

2.5 Structure for the realization of LTI systems

2.5 Structure for the realization of LTI systems

• Direct Form 2

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2.5 Structure for the realization of LTI systems

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