1. CHAPTER 2 Discrete-Time Signals and Systems in the Time-Domain
YANG Jian
School of Information Science and Technology
Yunnan University

2. Outline Discrete-Time Signals
Typical Sequences and Sequence Representation
The Sampling Process
Discrete-Time Systems
Time-Domain Characterization of LTI Discrete-Time Systems
Finite-Dimensional LTI Discrete-Time Systems
Correlation of Signals
Summary
云南大学滇池学院课程：现代信号处理数字信号处理

3. Discrete-Time Signals
Basic signals
Unit sample or unit impulse sequence
Unit step sequence
Exponential sequence
Signal classification
Continuous-time / discrete-time signals
Deterministic / random signals
Energy signals
signals with finite energy
Power signals
signals with finite power
Energy signals have zero power, and power signals have infinite energy
云南大学滇池学院课程：现代信号处理数字信号处理

4. Discrete-Time Signals
Time-Domain Representation
Sequence of numbers:
— sequence
— samples
— sample value or nth samples, a real or complex value
Figure of sequence:
is defined only for integer value of
云南大学滇池学院课程：现代信号处理数字信号处理

5. Discrete-Time Signals
Operation on sequences
Basic operation
Adder / Subtraction:
Scalar multiplication ( gain / attenuation ):
Delay / Advance:
Combination of Basic Operations
Multiplier:
Linear combination:
云南大学滇池学院课程：现代信号处理数字信号处理

6. Discrete-Time Signals
Operation on sequences
Sampling Rate Alteration ( special operations of for discrete-time signals )
Up-sampling:
Down-sampling:
云南大学滇池学院课程：现代信号处理数字信号处理

7. Discrete-Time Signals
Classification of Sequences
The number of sequences: finite / infinite
Finite-length sequences:
Symmetry
conjugate-symmetric ( even ):
conjugate-antisymmetric ( odd ):
Periodity: periodic / aperiodic
Periodic sequence:
云南大学滇池学院课程：现代信号处理数字信号处理

8. Discrete-Time Signals
Classification of Sequences
Energy and Power Signals
云南大学滇池学院课程：现代信号处理数字信号处理

9. Discrete-Time Signals
Classification of Sequences
Other types of Classification
Bounded:
Absolutely summable:
Square-summable:
云南大学滇池学院课程：现代信号处理数字信号处理

10. Typical Sequences and Sequence Representation
Some Basic Sequences
Unite sample sequence:
An arbitrary sequence can be represented by unite sample sequence in time-domain
Unite step sequence:
云南大学滇池学院课程：现代信号处理数字信号处理

11. Typical Sequences and Sequence Representation
Sinusoidal and Exponential Sequences
The real sinusoidal sequence:
The exponential sequence:
The sinusoidal sequence are periodic of period N as long as is an integer multiple of The smallest possible N is the fundamental period of the sequence.
云南大学滇池学院课程：现代信号处理数字信号处理

12. Typical Sequences and Sequence Representation
Some Typical Sequences
Regular window sequence:
Real exponential sequence:
Representation of an Arbitrary Sequence
An arbitrary sequence can be represented as a weight sum of basic sequence and its delayed version.
云南大学滇池学院课程：现代信号处理数字信号处理

13. The Sampling Process Uniform sampling:
Often the discrete-time sequence is developed by uniformly sampling a continuous-time signal :
the sampling
frequency
angular frequency
云南大学滇池学院课程：现代信号处理数字信号处理

14. The Sampling Process Aliasing:
When , a continuous-time sinusoidal signal of higher frequency would acquire the identity of a sinusoidal sequence of lower frequency after sampling.
e.g.
云南大学滇池学院课程：现代信号处理数字信号处理

15. Discrete-Time System Discrete-time system H [ ]
Output y(n)
Input x(n)
Simple Discrete-Time Systems
The accumulator
The M-point moving-average filter
The factor-of-L interpolator
云南大学滇池学院课程：现代信号处理数字信号处理

16. Discrete-Time System Classification of Discrete-Time System
Linear system:
Shift-Invariant System:
LTI System:
The linear time-invariable discrete-time system satisfies both the linear and the time-invariable properties.
云南大学滇池学院课程：现代信号处理数字信号处理

17. Discrete-Time System Classification of Discrete-Time System
Causal System:
In a causal discrete-time system, the th output sample
depends only on input samples for and
does notdepend on input samples for
云南大学滇池学院课程：现代信号处理数字信号处理

18. Discrete-Time System Classification of Discrete-Time System
Stable System:
Definition of bounded-input, bounded-output ( BIBO ) stable.
Passive and Lossless Systems
The passivity:
The losslessness: the same energy
云南大学滇池学院课程：现代信号处理数字信号处理

19. Discrete-Time System Impulse and Step Responses
Input sequence → output sequence
Impulse response :
Step response :
云南大学滇池学院课程：现代信号处理数字信号处理

20. Time-Domain Characterization of LTI Discrete-Time Systems
Input-Output Relationship
The response y(n) of the LTI discrete-time system to x(n) will be given by the convolution sum:
The operation
Step 1, time-reverse:
Step 2, shift n sampling period:
Step 3, product:
Step 4, summing all samples:
云南大学滇池学院课程：现代信号处理数字信号处理

21. Time-Domain Characterization of LTI Discrete-Time Systems
Some useful properties of the convolution operation
Commutative:
Associative for stable and single-sided sequences:
Distributive:
云南大学滇池学院课程：现代信号处理数字信号处理

22. Time-Domain Characterization of LTI Discrete-Time Systems
Simple Interconnection Schemes
Cascade Connection:
Parallel Connection:
Inverse System:
云南大学滇池学院课程：现代信号处理数字信号处理

23. Time-Domain Characterization of LTI Discrete-Time Systems
Stability Condition in Terms of the Impulse Response
An LTI digital filter is BIBO stable if only if its impulse response sequence is absolutely summable, i.e.:
Causality Condition in Terms of the Impulse Response
An LTI discrete-time system is causal if and only if its impulse response is a causal sequence satisfying the condition:
云南大学滇池学院课程：现代信号处理数字信号处理

24. Finite-Dimensional LTI Discrete-Time Systems
The difference equation:
An important subclass of LTI discrete-time systems is characterized by a linear constant coefficient difference equation of the form:
The order of the system is given by max( N, M )
云南大学滇池学院课程：现代信号处理数字信号处理

25. Finite-Dimensional LTI Discrete-Time Systems
Total Solution Calculation
The complementary solution
The homogeneous difference equation:
The characteristic equations:
云南大学滇池学院课程：现代信号处理数字信号处理

26. Finite-Dimensional LTI Discrete-Time Systems
Total Solution Calculation
The particular solution is of the same form as specified input
The total solution:
云南大学滇池学院课程：现代信号处理数字信号处理

27. Finite-Dimensional LTI Discrete-Time Systems
Zero-Input Response and Zero-State Response
zero-input response = complementary solution with initials;
zero-state response = the convolution sum of x(n) and h(n).
云南大学滇池学院课程：现代信号处理数字信号处理

28. Finite-Dimensional LTI Discrete-Time Systems
Impulse Response Calculation
云南大学滇池学院课程：现代信号处理数字信号处理

29. Finite-Dimensional LTI Discrete-Time Systems
Impulse Response Calculation
The solutions
云南大学滇池学院课程：现代信号处理数字信号处理

30. Finite-Dimensional LTI Discrete-Time Systems
Location of Roots of Characteristic Equation for BIBO Stability
A casual LTI system characteristic of a linear constant coefficient difference equation is BIBO stable, if the magnitude of each of the roots its characteristic equation is less than 1.
The necessary and sufficient condition:
云南大学滇池学院课程：现代信号处理数字信号处理

31. Finite-Dimensional LTI Discrete-Time Systems
Classification of LTI System
Based on impulse response length
Finite impulse response ( FIR ):
Infinite impulse response ( IIR ):
云南大学滇池学院课程：现代信号处理数字信号处理

32. Finite-Dimensional LTI Discrete-Time Systems
Classification of LTI System
Based on the output calculation process
Non-recursive system:
If the output sample can be calculated sequentially, knowing only the present and pass input samples.
Recursive system:
If the computation of the output involves past output samples.
Remarks:
FIR — Non-recursive
IIR — Recursive
云南大学滇池学院课程：现代信号处理数字信号处理

33. Correlation of Signals
Definitions
A measure of similarity between a pair of energy signals, x(n) and y(n), is given by the cross-correlation sequence defined by:
The autocorrelation sequence of x(n) is given by:
云南大学滇池学院课程：现代信号处理数字信号处理

34. Correlation of Signals
Properties of Autocorrelation and Cross-correlation Sequences
Set and as energies of the sequences x(n) and y(n) , then we can get or equivalently
If y(n) = x(n), then
The sample value of the autocorrelation sequence has its max value at zero lag ( l = 0 ).
云南大学滇池学院课程：现代信号处理数字信号处理

35. Correlation of Signals
Properties of Autocorrelation and Cross-correlation Sequences
If , where N is integer and b>0 is an arbitrary number. In this case , so
云南大学滇池学院课程：现代信号处理数字信号处理

36. Correlation of Signals
Normalized Forms of Correlation:
Correlation Computation for Power and Periodic Signals
Power signals:
Periodic signals:
云南大学滇池学院课程：现代信号处理数字信号处理

37. Summary The LTI system has numerous applications in practice.
The LTI system can be described by an input-output relation composed of a linear constant coefficient difference equation.
The LTI discrete-time system is usually classified in terms of the length of its impulse response.
云南大学滇池学院课程：现代信号处理数字信号处理