1. Linear Systems & Signals
Basic definitions
Examples of signals and signal processing
Classification of signal models
Time-invariant & Linear Systems (TILSs)
TILS transfer function
Components of a TILS response
TILS response to a harmonic input
Summary
„Signal Theory” Zdzisław Papir

2. Basic definitions SYSTEM Signal Theory is related to modeling of both:
input
signals
output
Signal – variation of some physical quantity in (t;x,y,z).
Input signals – signals driving the system.
Output signals – response of the system to input signals.
Signal Theory is related to modeling of both:
signal properties,
signal processing in systems.
Signal/system model – description of signal/system
using functions or differential/integral equations
„Signal Theory” Zdzisław Papir

3. Examples of signals & signal processing
INFORMATION TRANSMISSION:
radio and television signals,
mobile and fixed telephony
data transmission (data networks)
OBJECT IDENTIFICATION SIGNALS:
ultrasound scanning,
X-ray scanning,
radar techniques,
stock analysis,
demographic trends.
„Signal Theory” Zdzisław Papir

4. Types of models of signals & signal processing
Analog models Discret models
Time-invariant models
Time-variant models
Linear models
Nonlinear models
Lumped models Distributed models
Deterministic models
Stochastic models
Static models Dynamic models
„Signal Theory” Zdzisław Papir

5. Analog models In analog models input and output signals
are continuous functions of time.
Seismogram recorded on an analog device
Electrocardiogram recorded on an analog device
„Signal Theory” Zdzisław Papir

6. Discret models Buffer Transmission channel t
In discret models signals are changing stepwise.
Buffer
Transmission channel 3 t 1 2 4 5 6 7
Packet count is one of the
possible teletraffic models.
„Signal Theory” Zdzisław Papir

7. Static models Buffer Channel Static models do not depend on time.
Packet buffering leads to multiplexing
of traffic streams over a channel.
„Signal Theory” Zdzisław Papir

8. Dynamic models Buffer Channel Diffusion approximation
Dynamic models do depend on time.
Buffer
Channel
Diffusion approximation
„Signal Theory” Zdzisław Papir

9. Time – invariant models
In time-invariant models both signal parameters and system characteristics do not depend on time. IN
OUT
LOGISTIC
ITERATION
FEEDBACK
„Teoria sygnałów” Zdzisław Papir

10. Time-invariant models
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„Signal Theory” Zdzisław Papir

11. Time-variant models Frequency Modulation FM
In time-variant models both signal parameters and system characteristics do depend on time.
Frequency Modulation FM
Instantaneous frequency of the FM signal
depends on the modulating signal.
„Signal Theory” Zdzisław Papir

12. Linear models R C r x1(t) y1(t) x2(t) y2(t) Preemphasis filter
In linear models the system response to a composite input signal is combination of system responses to component signals. R C r
x1(t)
y1(t)
x2(t)
y2(t)
Preemphasis filter
„Signal Theory” Zdzisław Papir

13. Linear models H(f) [dB] f [dec] Preemphasis filter f2/f1 = 1001 2 3 4 -2 -1
f [dec]
H(f) [dB]
Preemphasis filter f2/f1 = 100
log-log amplitude response
„Signal Theory” Zdzisław Papir

14. Nonlinear models Weber-Fechner Law
In nonlinear models the system response to a composite input signal is not combination of system responses to component signals.
Weber-Fechner Law
The sensation change depends linearly on
a relative stimulus change.
„Signal Theory” Zdzisław Papir

15. Nonlinear models -compression y x
The aim of a nonlinear compression is to emphasize weak signals while leaving strong signals almost unchanged.
-compression
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„Signal Theory” Zdzisław Papir

16. Nonlinear models Kompresja 
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Kompresja 
Signal before compression
Signal after compression
-compression law is used in Northern America; European digital telephony exploits the A-compression concept.
„Teoria sygnałów” Zdzisław Papir

17. Lumped models
In lumped models energy is accumulated/disspated in isolated system points. Signals are transferred within the system without any delay. R C r
„Signal Theory” Zdzisław Papir

18. Distributed parameter models
In distributed models energy is accumulated/disspated in all system points. Signals are transferred within the system with some delay.
power networks
CATV coaxial network
Digital Subscriber Lines
Printed Circuit Boards (> 100 MHz)
„Signal Theory” Zdzisław Papir

19. Deterministic models In deterministic models signal fluctuations
are described by functions or equations. The exact formula modeling the signal makes future signal values known.
Double-sideband
Amplitude modulation AM
„Signal Theory” Zdzisław Papir

20. Stochastic models + – Transition graph for the Miller’s code
Stochastic models allow for a signal description exact to a probability distribution. The future signal values can be predicted with some accuracy only. + –
Transition graph for the Miller’s code
„Signal Theory” Zdzisław Papir

21. Miller’s code + – 1
„Signal Theory” Zdzisław Papir

22. Spectral density function
Miller’s code + – 
S()
Spectral density function
Bipolar code
Miller’s code
Biphase code
„Teoria sygnałów” Zdzisław Papir

23. Time-invariant Linear Systems (TILS)
„Signal Theory” Zdzisław Papir

24. Time-invariant System
Exponential input
TILS
Linear System
Time-invariant System
„Signal Theory” Zdzisław Papir

25. Exponential input TILS
The single and nontrivial solution to an equation:
is an exponential signal:
The amplitude H(s) depends on some constant s C.
The exponential signal is an invariant to Linear
Time-invariant Systems (TILS).
„Signal Theory” Zdzisław Papir

26. Exponential input TILS Let’s assume that an extra solution does exist:
Let’s substracte the identity side by side:
„Signal Theory” Zdzisław Papir

27. Exponential input We state that: The conclusion is:
We do not receive a new solution:
„Signal Theory” Zdzisław Papir

28. TILS transfer function
The transfer function of any TILS:
is defined as a ratio of the system response to
the exponential driving function. The transfer function can be interpreted as a TILS „amplification”.
„Signal Theory” Zdzisław Papir

29. TILS  (R, L, C) impedance TILS R C L
TILS impedance (voltage/current transfer function):
„Signal Theory” Zdzisław Papir

30. TILS  (R, L, C) admittance
ULS R C L
Admittance (current/voltage transfer function):
„Signal Theory” Zdzisław Papir

31. TILS  (R, L, C) transfer function
Derivation of the TILS  (R, L, C) transfer function is supported by various theorems:
serial/parallel combination of impedances,
Kirchoff’s current law,
Kirchoff’s voltage law,
Thevenin/Norton theorems,
transformation of current/voltage sources.
„Signal Theory” Zdzisław Papir

32. Preemphasis filter R
1/Cs r
x(t)
y(t)
„Signal Theory” Zdzisław Papir

33. TILS response to a sinusoidal input
TILS response to a sinusoidal (harmonic) input:
TILS
„Signal Theory” Zdzisław Papir

34. Harmonic excitation TILS
The transfer function H(j) is a rational function so it follows the Hermite symmetry:
Using the exponential representation we get:
„Signal Theory” Zdzisław Papir

35. Harmonic excitation TILS TILS response to the harmonic excitation:
A() - amplitude-frequency characteristic
() - phase -frequency characteristic
A-f function A() is an even function, A() = A(-)
P-f function () is an odd function, () = - (-)
„Signal Theory” Zdzisław Papir

36. Preemphasis filter R
1/Cs r
x(t)
y(t)
„Signal Theory” Zdzisław Papir

37. Preemphasis filter H(f) [dB] f [dek] Preemphasis filter f2/f1 = 1001 2 3 4 -2 -1
f [dek]
H(f) [dB]
Preemphasis filter f2/f1 = 100
Log-log amplitude response
„Signal Theory” Zdzisław Papir

38. Butterworth filter A-f function n = 2, fg = 1 kHz10 -2 2 4 6 -4
-200
-150
-100
-50
A-f function n = 2, fg = 1 kHz
P-f function n = 2, fg = 1 kHz
„Signal Theory” Zdzisław Papir

39. Butterworth filter
Butterworth filters have a maximaly flat a-f function in both passband and stopband.
„Signal Theory” Zdzisław Papir

40. Chebyshev filter Chebyshev polynomials:
Oscillation level of A2() in the passband:
The Chebyshev a-f function decreases faster than the Butterworth a-f function (for the same order).
„Signal Theory” Zdzisław Papir

41. Chebyshev filter Chebyshev Butterworth n = 6
„Signal Theory” Zdzisław Papir

42. Summary Signal Theory is related to modeling of both:
signal properties,
signal processing in systems.
In time-invariant models both signal parameters and system characteristics do not depend on time.
In linear models the system response to a composite input signal is combination of system responses to component signals.
The exponential signal is an invariant to Linear
Time-invariant Systems (TILS).
The transfer function of any TILS is defined as a ratio of the system response to the exponential driving function.
„Signal Theory” Zdzisław Papir

43. Summary
The transfer function of the TILS = (R, L, C) can be derived from a differential equation or using theorems of the circuit theory.
The TILS response to a harmonic excitation is a harmonic signal as well. The frequency remains unchanged. Amplitude and phase can be derived from amplitude and phase functions.
„Signal Theory” Zdzisław Papir