Signals The story is told that young King Solomon was given the choice between wealth and wisdom. When he chose wisdom, God was so pleased that he gave Solomon not only wisdom but wealth also. So it is with science. Arthur Holly Compton
What is a signal? Signals 1 What is a signal? A dictionary might define a signal as an electrical quantity or effect, as current, voltage, or electromagnetic waves, that can be varied in such a way as to convey information. Non-electromagnetic quantities such as temperature, pressure, or position can also convey information, but these are often converted to electromagnetic signals when measured and processed, so the dictionary definition is a good one for our purposes.
What is a signal? Signals 2 microphone voltage 0 0.1 0.2 0.3 time (seconds) A signal corresponding to the spoken word hello.
What is a signal? Signals 3 Signals in time: s(t) t is the independent variable; and s is the dependent variable. Signals in space: s(r), r = x y z Signals in space and time: s(r,t)
Graphical representation of signals Signals 4 Graphical representation of signals latitude longitude altitude time time time Graphical representation of three one-dimensional signals (latitude, longitude, altitude) that might represent the position of a jet as it flies between two cities.
Graphical representation of signals Signals 5 latitude Temperature, degrees Celsius longitude Graphical representation the surface temperture of the Great Lakes as a function of spatial position (latitude and longitude).
Graphical representation of signals Signals 6 vertical position 100 horizontal position 0 Graphical representation of a two-dimensional, scalar signal. Spatial position is the two-dimensional independent variable, and light intensity (brightness) is the dependent variable. This signal is a black-and-white photograph acquired with a digital camera.
Mathematical representation of signals Signals 7 Mathematical representation of signals exponential: s(t) =e αt, parameterized by α; third-order polynomial: s(t) =at 3 + bt 2 + ct + d, parameterized by a, b, c, and d; and sinusoidal signal: s(t) =A cos(2πft + φ), parameterized by A, f, and φ.
One-Dimensional Sinusoidal Signal in Time Signals 8 One-Dimensional Sinusoidal Signal in Time s(t) =A cos(2πft + φ) A is the amplitude; f is the frequency in Hz or cycles/s; and φ is the phase in radians. Notes: θ = φ + k2π is the same phase as φ. Do you see why? The period is T =1/f seconds.
One-Dimensional Sinusoidal Signal in Time Signals 9 5V x(t) 0 0.1 0.2 t -5V The one-dimensional sinusoidal signal with amplitude A =5V, frequency f =10Hz, and phase φ = π/2 rad.
One-Dimensional Sinusoidal Signal in Time Signals 10 x(t) 12 12 o clock 9 x(t) 3 3 o clock 9 o clock t 6 6 o clock The relationship between angular rotation around a clock and a sinusoidal signal.
One-Dimensional Sinusoidal Signal in Time Signals 11 Many students find it confusing to determine the phase of a sinusoidal signal from a graphical plot. Here is how it is done. First, find the positive peak that is nearest to the time origin (t =0), and determine the time at which this peak occurs. Call this time τ. The phase of the sinusoidal signal is then φ = 2πfτ. If you need to plot a sinusoidal signal, and you know its phase, then you can use this procedure in reverse. Just shift the peak of the cosine from the origin (t =0)tothetime τ = φ 2πf.
One-Dimensional Sinusoidal Signal in Time Signals 12 Example Determining the phase of a sinusoid A x(t) 0 2 16 t -A The phase for this signal is: φ = 2πfτ = 2π(1/16)2 = π/4
Complex-valued Signals Signals 13 Complex-valued Signals Complex-valued signals are a convenient way to keep track of two real-valued signals: x(t) =x R (t)+jx I (t). 1 x R (t) 1-1 0 2 3 4 t 0/4 x I (t) 1 0 x I (t) 1 3 4 t 3 2 1 x R (t) -1 2
Complex-valued Sinusoid Signals 14 Complex-valued Sinusoid x(t) =sin(2πft)+j cos(2πft) 1 x R (t) 1-1 0 2 3 4 t 0/4 x I (t) 1 0 x I (t) 1 3 4 t 3 2 1 x R (t) -1 2
Euler s Relationship Signals 15 Euler s Relationship Say oi ler! Ae jθ = A cos(θ)+jasin(θ) imaginary part Asin A Acos real part
Euler s Relationship Signals 16 Euler s Relationship = A cos(θ)+jasin(θ) + Ae jθ = A cos(θ) jasin(θ) A ( e jθ + e jθ) =2Acos(θ) Ae jθ cos(θ) = ejθ + e jθ 2
Euler s Relationship Signals 17 Euler s Relationship = A cos(θ)+jasin(θ) Ae jθ = A cos(θ)+jasin(θ) A ( e jθ e jθ) = j2a sin(θ) Ae jθ sin(θ) = ejθ e jθ 2j
Phasors Signals 18 Phasors x(t) = A cos(2πft + φ) = Re {Ae j(2πft+φ)} = Re { Ae jφ e j2πft} = Re { Xe j2πft}
Phasors Signals 19 Phasors Phasors are linear: α 1 x 1 (t) = A 1 cos(2πft + φ 1 ) + α 2 x 2 (t) = A 2 cos(2πft + φ 2 ) x(t) = Re { (α 1 X 1 + α 2 X 2 )e j2πft} X = α 1 X 1 + α 2 X 2