L1 Intro to Signals

Signal

any detectible quantity changing over time that conveys information

ex: noise, vibration, waves

Dimensionality

The dimension of the signal; used to describe signals

Continuous-time signal

s(t); signal value at all points; happens in nature (neurons)

Discrete-time signal

s[n]; takes samples very quickly over time

S[n] = s(nT); T = sampling period

Sampling

the reduction of a continuous time signal to a discrete time signal

Ideal C-to-D converter

Performs sampling and band-limited interpolation to convert continuous signals into discrete signals

A-to-D Converter

Performs sampling, quantization of sample, and encoding to change continuous analog signals to discrete digital signals

Amplitude

A; maximum value from equilibrium (0)

Radian frequency

omega nought; how quickly the sinusoid changes; rad/s

Phase shift

phi; change in timing of wave’s oscillations; radians

Cyclic frequency

f; how many cycles of the waveform per second; Hz or 1/s

Period

T; time for one cycle; seconds

Fourier analysis

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Even functions and sequences

Symmetric with respect to vertical axis; cosine

x(t) = x(-t), x[n] = x[-n]

Time shifting/translation: delayed and advanced in time

Shifting of signal along time axis

Delayed: b>0 y is shifted to the right by |b| relative to x

Advanced: b<0 y is shifted to the left by |b| relative to x

Time reversal/reflection

The new signal is the mirror image of the original signal about the y-axis

y(t) = x(-t)

Time compression/expansion or dilation

Reducing the amount of data required to represent a signal while preserving its essential characteristics

a>1 y is compressed along the horizontal axis by a factor of a

a<1 y is expanded along the horizontal axis by a factor of 1/a

Time scaling

the compression or expansion of a signal in time by changing the speed of signal flow.

y(t) = x(at)

|a|>1 the function is compressed along time axis by a factor of |a|

|a|<1 the function is expanded along time axis by a factor of |1/a|

a<0 the function is time reversed

Amplitude scaling

y(t) = ax(t)

Amplitude shifting

y(t) = x(t)+a

Real exponential function

.x(t) = Ae^(lambda t)

lambda>0 x(t) increases exponentially as t increases

lambda<0 x(t) decreases exponentially as t increases

lambda = 0 x(t) = A

Odd functions and sequences

Symmetric with respect to the origin; sine

x(t) = -x(-t), x[n] = -x[-n]