Discrete time signals

• Discrete time systems

• LTI systems

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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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 x(-
n) and dividing by 2.
– 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
• Block processing methods:
– Direct form
– Convolution table
– LTI form
– Matrix form
– Flip-and-slide form
– Overlap-add block convolution form
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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
– Consider a causal FIR filter of order M with impulse
response h(n), n = 0, 1, . . . , M. It may be represented as
a block:
– For the direct form
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Block processing
• Direct Form
– Consider the case of an order-3 filter and a length-5 input
signal.
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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 input signal
– X can be written by
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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
– The input is divided into the following three contiguous
blocks
– Convolving each block separately with h = [1, 2, −1, 1]
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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
in the interval 0 ≤ k ≤ n
– In the different way
– Hence
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Recursive system
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:
– yh(n): homogeneous solution
– yp(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
• Direct Form 1
– M+N delays
– M+N+1
multipliers
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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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