INTRODUCTION TO SIGNALS Chapter 2 INTRODUCTION TO SIGNALS DeSiaMore www.desiamore.com/ifm

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. DeSiaMore www.desiamore.com/ifm

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. DeSiaMore www.desiamore.com/ifm

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. DeSiaMore www.desiamore.com/ifm

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. DeSiaMore www.desiamore.com/ifm

Energy and Power DeSiaMore www.desiamore.com/ifm

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  ). DeSiaMore www.desiamore.com/ifm

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. DeSiaMore www.desiamore.com/ifm

Evaluate E and P and determine the type of the signal It is a Power signal 1 2 8 DeSiaMore www.desiamore.com/ifm

Evaluate E and P and determine the type of the signal It is an energy signal DeSiaMore www.desiamore.com/ifm

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. DeSiaMore www.desiamore.com/ifm

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. DeSiaMore www.desiamore.com/ifm

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). DeSiaMore www.desiamore.com/ifm

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). DeSiaMore www.desiamore.com/ifm

Given f(t), sketch 4–3f(–2t–6) -2 2 -1 6 f(t) DeSiaMore www.desiamore.com/ifm

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). DeSiaMore www.desiamore.com/ifm

Unit Impulse Function (Dirac delta function) Mathematical Definition: The unit impulse function (t) satisfies the following conditions: 1. (t) = 0 if t  0, 2. DeSiaMore www.desiamore.com/ifm

Properties of Delta Function f(t)d(t) = f(0)d(t) DeSiaMore www.desiamore.com/ifm

Trigonometric Fourier Series A signal g(t) in the interval t1  t  t1+T0 can be represented by T0 = 2 / 0 DeSiaMore www.desiamore.com/ifm

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 ; DeSiaMore www.desiamore.com/ifm

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. DeSiaMore www.desiamore.com/ifm

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. DeSiaMore www.desiamore.com/ifm

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. DeSiaMore www.desiamore.com/ifm