1. SIGNALS AND SYSTEMS AND INTRODUCTION TO DIGITAL SIGNAL PROCESSING
CSE, RU
Dr. Md. Ekramul Hamid Associate Professor Department of Computer Science and Engineering University of Rajshahi

2. Digital Signal Processing
Signal Processing deals with the enhancement, extraction, and representation of information for communication or analysis
Many different fields of engineering rely upon signal processing technology
Acoustics, telephony, radio, television, seismology, and radar are some examples
Initially, signal processing systems were implemented exclusively with analog hardware
However, recent advances in high-speed digital technology have made discrete signal processing systems more popular.
Digital systems have an advantage over analog systems in that they can process signals with an extraordinary degree of precision
Unlike the resistive and capacitive networks of analog systems, digital systems can be built numerically with the simple operations of addition and multiplication.
Digital Signal Processing is a field of numerical mathematics that is concerned with the processing of discrete signals
This area of mathematics deals with the principles that underlie all digital systems

3. Goal of DSP

4. Typical System Components

5. Applications of DSP

6. Applications of DSP: Multimedia
Compression: Fast, efficient, reliable transmission and storage of data
Applied on audio, image and video data for transmission over the Internet, storage
Examples: CDs, DVDs, MP3, MPEG4, JPEG
Mathematical Tools: Fourier Transform, Quantization, Modulation

7. Applications of DSP: Biological signal
Examples:
Brain signals (EEG)
Cardiac signals (ECG)
Medical images (x-ray, PET, MRI)
Goals:
Detect abnormal activity (heart attack)
Help physicians with diagnosis
Tools: Filtering, Fourier Transform

8. Applications of DSP: Biometrics
Identifying a person using physiological characteristics
Examples:
Fingerprint Identification
Face Recognition
Voice Recognition

9. Applications of DSP: Audio Signal processing
Active noise cancellation: Adaptive filtering
Headphones used in cockpits
Digital Audio Effects
Add special music effects such as delay, echo, reverb
Audio signal separation
Separate speech from interference

10. Main Topics to be Covered:
Signals And Systems – (Prerequisite of DSP)
Linear Time Invariant Systems
Convolution
Correlation

11. Signals and Systems Defined
A signal is any physical phenomenon which conveys information
Systems respond to signals and produce new signals
Excitation signals are applied at system inputs and response signals are produced at system outputs

12. Signal

13. Signal (Example)

14. Independent Variable
Time is often the independent variable for a signal. x(t) will be used to represent a signal that is a function of time, t.
A temporal signal is defined by the relationship of its amplitude (the dependent variable) to time (the independent variable).
An independent variable can be 1D (time), 2D (space), 3D (space) or even something more complicated.
The signal is described as a function of this variable.
There are many types of functions that can be used to describe signals (continuous, discrete, random).

15. Analog and Digital Signal
Signals can be analog or digital.
Analog signals can have an infinite number of values in a range.
Digital signals can have only a limited number of values.

16. Analog or continuous Time Signal
Most of the signals in the physical world are CT signals, since the time scale is infinitesimally fine (e.g., voltage, pressure, temperature, velocity).
Often, the only way we can view these signals is through a transducer, a device that converts a CT signal to an electrical signal.
Common transducers are the ears, the eyes, the nose… but these are a little complicated.
Simpler transducers are voltmeters, microphones, and pressure sensors.

17. Analog Signal Amplitude f(x) = 5 cos (x) Phase f(x) = 5 cos (x + 3.14)
Frequency
f(x) = 5 cos (x)
f(x) = 5 cos (x )
f(x) = 5 cos (3 x )

18. Signal in Time Domain The Independent Variable is Time
The Dependent Variable is the Amplitude
Most of the Information is Hidden in the Frequency Content
10 Hz
2 Hz
20 Hz
2 Hz +
10 Hz +
20Hz
Time
Magnitude

19. Analog to Digital Recording Chain
ADC
Microphone converts acoustic to electrical energy. It’s a transducer.
Continuously varying electrical energy is an analog of the sound pressure wave.
ADC (Analog to Digital Converter) converts analog to digital electrical signal.
Digital signal transmits binary numbers.
DAC (Digital to Analog Converter) converts digital signal in computer to analog for your headphones.

20. ADC: Step1: Sampling
Instantaneous amplitudes of continuous analog signal, measured at equally spaced points in time.
A series of “snapshots”

21. Analog to Digital Overview
Sampling Rate
How often analog signal is measured
[samples per second, Hz]
Example: 44,100 Hz
Sampling Resolution
[a.k.a. “sample word length,” “bit depth”] Precision of numbers used for measurement: the more bits, the higher the resolution.
Example: 16 bit

22. Sampling Rate Nyquist Theorem:
Determines the highest frequency that you can represent with a digital signal.
Nyquist Theorem:
Sampling rate must be at least twice as high as the highest frequency you want to represent.
Capturing just the crest and trough of a sine wave will represent the wave exactly.

23. Recovery of a sampled sine wave for different sampling rates

24. Aliasing What happens if sampling rate not high enough?
A high frequency signal
sampled at too low a rate
looks like …
… a lower frequency signal.
That’s called aliasing or foldover. An ADC has a low-pass anti-aliasing filter to prevent this.
Synthesis software can cause aliasing.

25. Anti-Aliasing Filter
An anti-aliasing filter removes frequencies that are higher than half the sampling rate using what is called a low pass filter. A low pass filter lets the low frequencies “pass" and "cut" the high frequencies. Low pass filters are sometimes called high cut filters.

26. Effects of under sampling

27. Effects of required sampling

28. Common Sampling Rates Most software can handle all these rates.
Which rates can represent the range of frequencies audible by (fresh) ears?
Sampling Rate
Uses
44.1 kHz (44100)
CD, DAT
48 kHz (48000)
DAT, DV, DVD-Video
96 kHz (96000)
DVD-Audio
22.05 kHz (22050)
Old samplers
Most software can handle all these rates.

29. A/D: Step2: Quantization
Sampling results in a series of pulses of varying amplitude values ranging between two limits: a min and a max.
The amplitude values are infinite between the two limits.
We need to map the infinite amplitude values onto a finite set of known values.
This is achieved by dividing the distance between min and max into L zones, each of height 
 = (max - min)/L

30. Quantization Level
The midpoint of each zone is assigned a value from 0 to L-1 (resulting in L values)
Each sample falling in a zone is then approximated to the value of the midpoint.

31. Assigning Codes to Zones
Each zone is then assigned a binary code.
The number of bits required to encode the zones, or the number of bits per sample as it is commonly referred to, is obtained as follows:
nb = log2 L
Given our example, nb = 3
The 8 zone (or level) codes are therefore: 000, 001, 010, 011, 100, 101, 110, and 111
Assigning codes to zones:
000 will refer to zone -20 to -15
001 to zone -15 to -10, etc.

32. Time — measure amp. at each tick of sample clock
3-bit Quantization
A 3-bit binary (base 2) number has 23 = 8 values. 1 2 3 4 5 6 7
Amplitude
Time — measure amp. at each tick of sample clock

33. 4-bit Quantization A 4-bit binary number has 24 = 16 values.2 4 6 8 10 12 14
Amplitude
Time — measure amp. at each tick of sample clock
A better approximation

34. Quantization Error
When a signal is quantized, we introduce an error - the coded signal is an approximation of the actual amplitude value.
The difference between actual and coded value (midpoint) is referred to as the quantization error.
The more zones, the smaller  which results in smaller errors.
BUT, the more zones the more bits required to encode the samples -> higher bit rate

35. Quantization Noise Random errors: sounds like low-amplitude noise
Round-off error: difference between actual signal and quantization to integer values…
Random errors: sounds like low-amplitude noise

36. A/D: step3: Coding Quantization
Quantization is the process of converting the sampled analog voltages into digital words.
Data coding
Data coding separates the digital words so that they are more easily identified.

37. Digital to Analog Conversion: Sample and Hold
To reconstruct analog signal, hold each sample value for one clock tick; convert it to steady voltage. 1 2 3 4 5 6 7
Amplitude
Time

38. DAC: Smoothing Filter
Apply an analog low-pass filter to the output of the sample-and-hold unit: averages “stair steps” into a smooth curve. 1 2 3 4 5 6 7
Amplitude
Time

39. Discrete-Time Signal (Example)
Discrete-time signals are represented by sequence of numbers
The nth number in the sequence is represented with x[n]
Often times sequences are obtained by sampling of continuous-time signals
In this case x[n] is value of the analog signal at xc(nT)
Where T is the sampling period 20 40 60 80
100
-10 10
t (ms) 30 50
n (samples)

40. Signals With Symmetry: Periodic
The periodicity of sequence if
: any integer
: positive integer
then the is called a periodic sequence,
and the value of N is called the fundamental period.

41. periodic sequence

42. Signals With Symmetry: Even/Odd

43. Signals With Symmetry: Even/Odd
Any signals can be expressed as a sum of even and odd signals. That is:
This is demonstrated to the right for a signal referred to as a unit step.

44. Real and Complex Signals
Complex signals are an important abstraction in many disciplines such as communications and multidimensional signal processing.
In general, x is a complex quantity and has:
a real and imaginary part, or equivalently
a magnitude and a phase angle.
A very important class of signals is complex exponentials:
CT signals of the form x(t) = est
DT signals of the form x[n] = zn
where z and s are complex numbers.
For example, suppose s = j/8 and z = e j/8, the exponentials are purely imaginary, then the real parts are:

45. Real and Complex Signals
For example, suppose s = σ + jω for a CT signal:
>0:
<0:
>0:
<0:
For example, suppose z = e(σ + jω) for a DT signal:

46. The Discrete-Time Signal: Sequences
Unit sample sequence
Unit step sequence
-10 -5 5 10
0.5 1
1.5
Exponential sequence

47. The Discrete-Time Signal: Sequences

48. The Discrete-Time Signal: Sequences

49. The Discrete-Time Signal: Sequences
Operations on sequence
Time-shifting operation
where is an integer
delaying operation
z-1
Unit delay
advance operation z
Unit advance

50. The Discrete-Time Signal: Sequences
Time-shifting operation
Time-shifting operation

51. The Discrete-Time Signal: Sequences
Time-reversal (folding) operation
Addition operation
Sample-by-sample addition
Adder

52. The Discrete-Time Signal: Sequences
folding operation
folding operation

53. addition operation

54. The Discrete-Time Signal: Sequences
Scaling operation
Multiplier A
Product (modulation) operation
Sample-by-sample multiplication
modulator

55. modulation operation

56. Decimation and Interpolation
Decimation---down-sampling N
x(n)

57. Decimation and Interpolation
Decimation---down-sampling N

58. Decimation and Interpolation
Decimation---down-sampling N
y(m)

59. Decimation and Interpolation
Interpolation --- up-sampling N

60. Decimation and Interpolation
Interpolation --- up-sampling N

61. The Discrete-Time Signal: Sequences
Sinusoidal sequence
amplitude
digital angular frequency
phase

62. The Discrete-Time Signal: Sequences

63. The Discrete-Time Signal: Sequences
The periodicity of sinusoidal sequence If
, : any integer
is a periodic sequence and its period is

64. The Discrete-Time Signal: Sequences
If is a integer
If is a noninteger rational number
If is a irrational number
is an aperiodic sequence

65. Periodicity of sequence

66. The Discrete-Time Signal: Sequencesns
Complex-valued exponential sequence
Attenuation factor

67. Complex-valued exponential sequence

68. The Discrete-Time System
Introduction
A discrete-time system processes a given input sequence x(n) to generate an output sequence y(n) with more desirable properties.
Mathematically, an operation T [ • ] is used.
y(n) = T [ x(n) ]
x(n): excitation, input signal
y(n): response, output signal

69. Example: Accumulator
The output at time instant n is the sum of the input sample at time instant n and the previous output at time instant n-1, which is the sum of all previous input sample values from to n-1
The input-output relation can also be written in the form:
This form is used for a causal input sequence, in which case y(-1) is called the initial condition

70. The Discrete-Time System
Classification
Linear System
Time-Invariant (Shift-Invariant) System
Linear Time-Invariant (LTI) System
Causal System
Stable System
Memory System

71. The Discrete-Time System
Linear System
A system is called linear if it has two mathematical properties: homogeneity and additivity.
Accumulator

72. The Discrete-Time System
Time-Invariant (Shift-Invariant) System
Accumulator

73. The Discrete-Time System
Linear Time-Invariant (LTI) System
A system satisfying both the linearity and the time-invariance properties is called an LTI system.
LTI systems are mathematically easy to analyze and characterize, and consequently, easy to design.

74. The Discrete-Time System
The output of an LTI system is called
linear convolution sum
An LTI system is completely characterized in the time domain by the impulse response h(n).

75. return

76. The Discrete-Time System
Causal System
In a causal system, the -th output sample depends only on input samples for and does not depend on input samples for
e.g.
For a causal system, changes in output samples do not precede changes in the input samples.

77. The Discrete-Time System
An LTI system will be a causal system if and only if :
An ideal low-pass filter is not a causal system !
A sequence is called a causal sequence if :

78. The Discrete-Time System
Stable System
A system is said to be bounded-input bounded-output (BIBO) stable if every bounded input produces a bounded output, i.e.
An LTI system will be a stable system if and only if :

79. The Discrete-Time System
Memory:
A system is memoryless if y[n] = f ( x[n] )
i.e. it sees only present values.
A system has memory if y [n] depends on previous values
it can also depend on present and future values!

80. DT Unit-Impulse Response
Consider the DT SISO system:
If the input signal is and the system has no energy at , the output is called the impulse response of the system
System
System

81. Example Consider the DT system described by
Its impulse response can be found to be

82. Linear Time-Invariant Systems and Convolution

83. Linear Time-Invariant Systems and Convolution

84. Linear Time-Invariant Systems and Convolution

85. Linear Time-Invariant Systems and Convolution

86. Linear Time-Invariant Systems and Convolution

87. Linear Time-Invariant Systems and Convolution

88. Correlation
Correlation addresses the question: “to what degree is signal A similar to signal B.”
An intuitive answer can be developed by comparing deterministic signals with stochastic signals.
Deterministic = a predictable signal equivalent to that produced by a mathematical function
Stochastic = an unpredictable signal equivalent to that produced by a random process

89. Correlation Correlation is maximum when
Two signals are similar in shape
And are in phase (or unshifted)
Correlation is measure of similarity between two signals as a function of time shift between them

90. Correlation
Correlation functions shows how similar two signals are, and how long they remain similar when one is shifted with respect to the other

91. Autocorrelation
Correlating a signal with itself

92. Autocorrelation
Autocorrelation can be used to extract a signal from noise

93. Autocorrelation to locate a signal
Cross correlation can be used to detect and locate known reference signal in noise

94. Xcorrelation to identify a signal
Cross correlation can be used to identify a signal by comparison with a library of known reference signal

95. Convolution and Correlation
R[n]= x[k]y[n+k] Vs. C[n] = x[k]y[n-k]

96. Convolution and Correlation
If one signal is symmetric, convolution and correlation are identical

97. Thanks for your attention