Elementary Signal Functions2020-04-12
The core functions used in DSP are:
These functions can be viewed as pulses or through windows.
Unit Impulse is a function that does not follow the strict rules of mathematics. By setting the integration of some (well-behaved, continuous at 0) function over some interval equal to 1, we establish the initial approximation of the Unit Impulse: A narrow pulse with unitary area.
By further narrowing the interval but maintaining the unit area, we require greater values from the original function. The Unit Impulse function is defined as the limit of this function as the interval width approaches zero. Therefore the Unit Impulse approaches infinity, but strictly speaking, the conditions of infinite amplitude and zero pulse width would not multiply to unit area -- but to undefined. That's why Unit Impulse is not, speaking strict mathematics, a true function.
But it's useful for our purposes to take that approximation. We say:
δ(t) = 0 where t != 0
δ(t) = inf where t = 0
The derivative of the Unit Impulse is the Unit Step. As we restrict the total area under
Derivations from different functions
x(t) = 1/T where |t| < T/2
x(t) = 0 where |t| >= T/2
δ(t) = limit of x(t) as T->0
x(t) = (1/2τ)e^(-|t|/τ) where -inf < t < inf
δ(t) = limit of x(t) as τ->0
x(t) = (1/σ√(2π))e^(-x^2/2σ^2) where -inf < t < inf
δ(t) = limit of x(t) as σ->0
x(t) = sin(πt/T)/(πt/T) where -inf < t < inf δ(t) = limit of x(t) as T->0
The Sieving definition is a more rigorous way to define Unit Impulse. Given a function f(t), well-behaved and continous at t0,
[-inf to inf] ∫ f(t) δ(t - t0) dt = f(t0)
The Unit Step function is akin to flipping on a switch -- it is zero for all t<0, and one for all t>=0:
u(t) = 1 where t >= 0
u(t) = 0 where t < 0
Unit Step is the integral of Unit Impulse. This follows from requiring the area under the Unit Impulse 'curve' be 1.
Integrating a function from zero to one (ignoring the paradoxical infinite rate of change) requires
As with any other function, delaying t (subtracting a const.) moves the 'switching' 'out' i.e. more positive on the time axis, so later in time. Advancing t (adding a constant) moves the switch 'in' i.e. more negative on the time axis, or sooner in time.
The Unit Ramp is zero for all t<0, and thereafter rises with a slope of one.
r(t) = t where t >= 0
r(t) = 0 where t < 0
Sinusoids are periodic functions, but combining them may result in aperiodic functions.
x(t) = V0 e^(αt)
The Exponential function grows or decays at a rate proportional to α. With +α, the exponential grows; with -α, the function decays. The Exponential function is unique for retaining its essential form throughout derivation and integration: only the magnitude changes, by
Exponentials may be complex: of the form e^(st), where s = σ + jω.