1. INTRODUCTION TO SIGNALS
Chapter 2
INTRODUCTION TO SIGNALS
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2. Definition
A Signal: is a function that specifies how a specific variable changes versus an independent variable such as time. Usually represented as an X-Y plot.
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3. Classification of Signals (1/4)
Analog vs. Digital signals:
Analog signals are signals with magnitudes that may take any value in a specific range.
Digital signals have amplitudes that take only a finite number of values.
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4. Classification of Signals (2/4)
Continuous-time vs. discrete-time: Continuous-time signals have their magnitudes defined for all values of t. They may be analog or digital. Discrete-time signals have their magnitudes defined at specific instants of time only. They may be analog or digital.
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5. Classification of Signals (3/4)
Periodic vs. aperiodic signals: Periodic signals are signals constructed from a shape that repeats itself regularly after a specific amount of time T0, that is: f(t) = f(t+nT0) for all integer n Aperiodic signals do not repeat regularly.
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6. Energy and Power
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7. Classification of Signals (4/4)
Energy Signals: an energy signal is a signal with finite energy and zero average power (0 ≤ E < , P = 0)
Power Signals: a power signal is a signal with infinite energy but finite average power (0 < P < , E  ).
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8. More on Energy and Power Signals
A signal cannot be both an energy and power signal.
A signal may be neither energy nor power signal.
All periodic signals are power signals (but not all non–periodic signals are energy signals).
Any signal f that has limited amplitude (|f| < ) and is time limited (f = 0 for |t |> t0) is an energy signal.
The square root of the average power of a power signal is called the RMS value.
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9. Evaluate E and P and determine the type of the signal
It is a Power signal 1 2 8
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10. Evaluate E and P and determine the type of the signal
It is an energy signal
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11. Basic Signal Operations (1/4)
Time Shifting: given the signal f(t), the signal f(t–t0) is a time-shifted version of f(t) that is shifted to the left if t0 is positive and to the right if t0 is negative.
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12. Basic Signal Operations (2/4)
Magnitude Shifting: Given the signal f(t), the signal c +f(t) is a magnitude-shifted version of f(t) that is shifted up if c is positive and shifted down if c is negative.
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13. Basic Signal Operations (3/4)
Time Scaling and Time Inversion: Given f(t), the signal f(at) is a time-scaled version of f(t), where a is a constant, such that f(at) is an expanded version of f(t) if 0<|a|<1, and f(at) is a compressed version of f(t) if |a|>1. If a is negative, the signal f(at) is also a time-inverted version of f(t).
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14. Basic Signal Operations (4/4)
Magnitude Scaling and Mag. Inversion: Given f(t), the signal bf(t) is a magnitude-scaled version of f(t), where b is a constant, such that bf(t) is an attenuated version of f(t) if 0<|b|<1, and bf(t) is an amplified version of f(t) if |b|>1. If b is negative, the signal bf(t) is also a magnitude-flipped version of f(t).
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15. Given f(t), sketch 4–3f(–2t–6)-2 2 -1 6
f(t)
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16. Unit Impulse Function (Dirac delta function)
Graphical Definition:
The rectangular pulse shape approaches the unit impulse function as  approaches 0 (notice that the area under the curve is always equal to 1).
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17. Unit Impulse Function (Dirac delta function)
Mathematical Definition:
The unit impulse function (t) satisfies the following conditions:
1. (t) = if t  0, 2.
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18. Properties of Delta Function
f(t)d(t) = f(0)d(t)
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19. Trigonometric Fourier Series
A signal g(t) in the interval t1  t  t1+T0 can be represented by
T0 = 2 / 0
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20. Or, in the compact form If g(t) is even then bn = 0 for all n
If g(t) is odd then an=0 for all n.
C0 = a0 ;
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21. Remarks on Fourier Series (FS) Representations
The frequency 0= 2p/T0 is called the fundamental frequency and the multiple of this frequency n0 is called the nth harmonic.
FS of g(t) is equal to g(t) over the interval t1  t  t1+T0 only.
The FS for all t is a periodic function of period T0 in which the segment of g(t) over the interval t1  t  t1+T0 repeats periodically.
If the function g(t) itself is periodic with period T0 then the FS represents g(t) for all t.
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22. Exponential Fourier Series
Dn is related to Cn and n as
| Dn | is called the amplitude spectrum of the signal.
 Dn is called the phase spectrum of the signal.
They provide a frequency-domain representation of the signal.
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23. Parseval’s Theorem
Let g(t) be a periodic signal. The power of g(t) is equal to the sum of the powers of its Fourier Components.
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