0. Chapter 2: Discrete-time signals and systems
Department of Computer Eng.
Sharif University of Technology
Discrete-time signal processing
Chapter 2: Discrete-time signals and systems
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer and Buck, © Prentice Hall Inc.

1. Chapter 2: Discrete time signals and systems
2.0 DSP: applications
Speech, audio
Noise reduction (Dolby), compression (MP3), …
Radar
filtering, movement detection, …
Image processing
Compression, pattern recognition, segmentation,…
Biomedical
Monitoring, analysis, tele-medicine, …
Chapter 2: Discrete time signals and systems 1

2. Chapter 2: Discrete time signals and systems
Signal Types
Signals
Continuous-time
Discrete-time
Continuous-value
Continuous-value
Discrete-value
Analog
Discrete
Digital
Chapter 2: Discrete time signals and systems 2

3. Chapter 2: Discrete time signals and systems
Signal Types
Analog signals: continuous in time and amplitude
Example: voltage, current, temperature,…
Digital signals: discrete both in time and amplitude
Example: attendance of this class, digitizes analog signals,…
Discrete-time signals: discrete in time, continuous in amplitude
Example: hourly change of temperature
Theory of digital signals would be too complicated
Requires inclusion of nonlinearities into theory
Theory is based on discrete-time continuous-amplitude signals
Most convenient to develop theory
Good enough approximation to practice with some care
In practice we mostly process digital signals on processors
Need to take into account finite precision effects
Chapter 2: Discrete time signals and systems 3

4. Chapter 2: Discrete time signals and systems
Signal Types
Continuous time –
Continuous amplitude
Discrete amplitude
Discrete time –
Chapter 2: Discrete time signals and systems 4

5. 2.1 Basic sequences and sequence operations
Delaying (Shifting) a sequence
Unit sample (impulse) sequence
Unit step sequence
Exponential sequences
Chapter 2: Discrete time signals and systems 5

6. Chapter 2: Discrete time signals and systems
Sinusoidal Sequences
Important class of sequences
An exponential sequence with complex
There are two important differences between continues-time and discrete-time sinusoids: in the discrete sinusoids
1-sinusoids with frequencies where k is an integer, are indistinguishable from one another.
2-is not necessary periodic with 2/o
Chapter 2: Discrete time signals and systems 6

7. 2.2 Discrete-Time Systems
A Discrete-Time System is a mathematical operation that maps a given input sequence x[n] into an output sequence y[n]
Example:
Moving (Running) Average
Maximum
Ideal Delay System
Chapter 2: Discrete time signals and systems 7

8. Chapter 2: Discrete time signals and systems
Memoryless System
A system is memoryless if the output y[n] at every value of n depends only on the input x[n] at the same value of n
Example :
Square
Sign
counter example:
Ideal Delay System
Chapter 2: Discrete time signals and systems 8

9. Chapter 2: Discrete time signals and systems
Linear Systems
Linear System: A system is linear if and only if
Example: Ideal Delay System
Chapter 2: Discrete time signals and systems 9

10. Time-Invariant Systems
Time-Invariant (shift-invariant) Systems
A time shift at the input causes corresponding time-shift at output
Example: Square
Counter Example: Compressor System
Chapter 2: Discrete time signals and systems 10

11. Chapter 2: Discrete time signals and systems
Causal System
A system is causal iff it’s output is a function of only the current and previous samples
Examples: Backward Difference
Counter Example: Forward Difference
Chapter 2: Discrete time signals and systems 11

12. Chapter 2: Discrete time signals and systems
Stable System
Stability (in the sense of bounded-input bounded-output BIBO). A system is stable iff every bounded input produces a bounded output
Example: Square
Counter Example: Log
Chapter 2: Discrete time signals and systems 12

13. Chapter 2: Discrete time signals and systems
LTI System Example
Chapter 2: Discrete time signals and systems 13

14. 2.3 Linear Time-Invariant Systems
Special importance for their mathematical tractability
Most signal processing applications involve LTI systems
LTI system can be completely characterized by their impulse response
Chapter 2: Discrete time signals and systems 14

15. Digital Signal Processing

16. Digital Signal Processing
Ex)
Digital Signal Processing

17. Digital Signal Processing

18. 2.4 Properties of LTI Systems
Convolution is commutative
Convolution is distributive
Chapter 2: Discrete time signals and systems 18

19. Properties of LTI Systems
Cascade connection of LTI systems
Chapter 2: Discrete time signals and systems 19

20. Stable and Causal LTI Systems
An LTI system is (BIBO) stable iff Impulse response is absolute summable
Let’s write the output of the system as
Then the output is bounded by
The output is bounded if the absolute sum is finite
An LTI system is causal iff
Chapter 2: Discrete time signals and systems 20

21. Digital Signal Processing
Stability Condition : A linear time-invariant system is stable If and only if
Digital Signal Processing

22. Digital Signal Processing
Causality Condition :
Neither necessary nor sufficient
condition for all systems,
But necessary and sufficient for LTI system
But x[n-k] for k>=0 shows
The future values of x[n].
So y[n] depends only on the
Future values of x[n].
Digital Signal Processing

23. 2.5 Linear Constant-Coefficient Difference Equations
An important class of LTI systems of the form
The output is not uniquely specified for a given input
The initial conditions are required
Linearity, time invariance, and causality depend on the initial conditions
If initial conditions are assumed to be zero system is linear, time invariant, and causal
Example
Moving Average
Chapter 2: Discrete time signals and systems 23

24. Digital Signal Processing
Linear Constant-Coefficient Difference Equations
Ex)
X[n] is the difference of y[n]
Digital Signal Processing

25. Digital Signal Processing

26. Digital Signal Processing
Frequency-Domain Representation
Digital Signal Processing

27. Digital Signal Processing
Ex)
Digital Signal Processing

28. Digital Signal Processing

29. Digital Signal Processing
Ex)
Digital Signal Processing

30. Eigenfunctions of LTI Systems
Complex exponentials are eigenfunctions of LTI systems:
Let’s see what happens if we feed x[n] into an LTI system:
The eigenvalue is called the frequency response of the system
is a complex function of frequency
Eigenfunction
Eigenvalue
Chapter 2: Discrete time signals and systems 30

31. 2.6&2.7 Discrete-Time Fourier Transform
Many sequences can be expressed as a weighted sum of complex exponentials as
Where the weighting is determined as
is the Fourier spectrum of the sequence x[n]
The phase wraps at 2 hence is not uniquely specified
The frequency response of a LTI system is the DTFT of the impulse response
Chapter 2: Discrete time signals and systems 31

32. Absolute and Square Summability
For a given sequence if the infinite sum convergence, the DTFT exist
All stable systems are absolute summable and have finite and continues frequency response
Chapter 2: Discrete time signals and systems 32

33. Absolute and Square Summability
Absolute summability is sufficient condition for DTFT
Some sequences may not be absolute summable but only square summable
Such sequences can be represented by fourier transform if
In other words, the error may not approach zero at each value of as but the total energy in the error does.
Chapter 2: Discrete time signals and systems 33

34. Example: Ideal Lowpass Filter
The periodic DTFT of the ideal lowpass filter is
The inverse can be written as
Not causal, Not absolute summable but it has a DTFT, The DTFT converges in the mean-squared sense
Role of Gibbs phenomenon
Chapter 2: Discrete time signals and systems 34

35. Digital Signal Processing
Ex)
The impulse response is not causal,
Not absolutely summable, but squarely summable,
Since sequence values approach zero as n-> infinity,
But only as 1/n
Digital Signal Processing

36. Role of Gibbs phenomenon
Chapter 2: Discrete time signals and systems 36

37. Digital Signal Processing
Is absolute summabality a necessary condition?
Consider the Fourier Transform:
absolute summabality is not a necessary condition.
Digital Signal Processing

38. Digital Signal Processing

39. Example: Generalized DTFT
DTFT of
Not absolute summable, Not even square summable But we define its DTFT as a pulse train
Let’s place into inverse DTFT equation
Chapter 2: Discrete time signals and systems 39

40. 2.8 Symmetric Sequence and Functions
Conjugate-symmetric
Conjugate-antisymmetric
Sequence
Function
Chapter 2: Discrete time signals and systems 40

41. Chapter 2: Discrete time signals and systems
Properties of DTFT
Sequence x[n]
Discrete-Time Fourier Transform X(ej)
x*[n]
X*(e-j)
x*[-n]
X*(ej)
Re{x[n]}
Xe(ej) (conjugate-symmetric part)
jIm{x[n]}
Xo(ej) (conjugate-antisymmetric part)
xe[n]
XR(ej)= Re{X(ej)}
xo[n]
jXI(ej)= jIm{X(ej)}
Any real x[n]
X(ej)=X*(e-j) (conjugate symmetric)
XR(ej)=XR(e-j) (real part is even)
XI(ej)=-XI(e-j) (imaginary part is odd)
|X(ej)|=|X(e-j)| (magnitude is even)
X(ej)=-X(e-j) (phase is odd)
XR(ej)
jXI(ej)
Chapter 2: Discrete time signals and systems 41

42. Example: Illustration of Symmetry Properties
DTFT of the real sequence x[n]=anu[n]
Some properties are
Chapter 2: Discrete time signals and systems 42

43. 2.9 Fourier Transform Theorems
x[n]
y[n]
X(ej)
Y(ej)
ax[n]+by[n]
aX(ej)+bY(ej)
x[n-nd]
x[-n]
X(e-j)
nx[n]
x[n]y[n]
X(ej)Y(ej)
x[n]y[n]
Chapter 2: Discrete time signals and systems 43

44. Fourier Transform Pairs
Sequence
DTFT
[n-no]
u[n]
cos(on+)
Chapter 2: Discrete time signals and systems 44

45. Chapter 2: Discrete time signals and systems
Example:
Determining an inverse fourier transform
Chapter 2: Discrete time signals and systems 45

46. Chapter 2: Discrete time signals and systems
Example:
Determining the Impulse response from the frequency response
Chapter 2: Discrete time signals and systems 46

47. Chapter 2: Discrete time signals and systems
Example:
Determining the Impulse response for a Difference Equation
To find the impulse response h[n], we set
Applying the DTFT to both sides of equation. We obtain
Chapter 2: Discrete time signals and systems 47

48. Example: