What is Unit step function: Definition and 55 Discussions

The Heaviside step function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or 𝟙), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one.
The function was originally developed in operational calculus for the solution of differential equations, where it represents a signal that switches on at a specified time and stays switched on indefinitely. Oliver Heaviside, who developed the operational calculus as a tool in the analysis of telegraphic communications, represented the function as 1.
The Heaviside function may be defined as:

a piecewise function:
an indicator function:
the derivative of the ramp function: The Dirac delta function is the derivative of the Heaviside function

Hence the Heaviside function can be considered to be the integral of the Dirac delta function. This is sometimes written as

although this expansion may not hold (or even make sense) for x = 0, depending on which formalism one uses to give meaning to integrals involving δ. In this context, the Heaviside function is the cumulative distribution function of a random variable which is almost surely 0. (See constant random variable.)
In operational calculus, useful answers seldom depend on which value is used for H(0), since H is mostly used as a distribution. However, the choice may have some important consequences in functional analysis and game theory, where more general forms of continuity are considered. Some common choices can be seen below.
Approximations to the Heaviside step function are of use in biochemistry and neuroscience, where logistic approximations of step functions (such as the Hill and the Michaelis–Menten equations) may be used to approximate binary cellular switches in response to chemical signals.

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  1. F

    I How to invert unit step function?

    I am trying to find out how to reverse the unit step function. The closest I could find is this sentence, which is more like a definition? "if we want to reverse the unit step function, we can flip it around the y-axis as such: u(-t). With a little bit of manipulation, we can come to an...
  2. kostoglotov

    An infinite sum of the Heaviside function

    I'm not sure where to put this question. It is by itself pretty basic, but it's a preamble to a Laplace Transform exercise, and I'll probably want to ask some follow up questions once the current query is resolved. 1. Homework Statement Unit stair-case function: f(t) = n, \ if \ \ n-1 \leq t...
  3. E

    Fourier Transform of Heaviside function

    Homework Statement Find the Fourier transform of H(x-a)e^{-bx}, where H(x) is the Heaviside function. Homework Equations \mathcal{F}[f(t)]=\frac{1}{2 \pi} \int_{- \infty}^{\infty} f(t) \cdot e^{-i \omega t} dt Convolution theory equations that might be relevant: \mathcal{F}[f(t) \cdot...
  4. J

    Laplace Transform unit step function

    Homework Statement g(t) = (t+1)us(t) - (t-1)us(t-1) - 2us(t-1) - (t - 2)us(t-2) + (t-3)us)(t - 3) + us(t-3) Homework Equations unit step function us(t-3) is same as u3 (t) Shift in time: L[f(t - T)us(t-T)] = e-TsF(s) us(t) ↔ 1/s t ⇔ 1/s2 The Attempt at a Solution 1/s2 + 1/s - e-s/s2 + e-s/s -...
  5. S

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    Homework Statement Show that δ(x-x') = d/dx Θ(x-x') Homework Equations ∫ f(x') δ(x-x') dx' = f(x) Θ(x-x') vanishes if x-x' is negative and 1 if x-x' is positive The Attempt at a Solution I saw a relation of the δ function but I don't know why is it like that. Integral of δ(x-x') from -∞ to x...
  6. C

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  7. J

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  8. sinaphysics

    Help me to survive from a big dilemma(Unit step function)

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  9. T

    Integral of Unit step function

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  10. U

    Evaluating and integral including unit step function

    how can this integral be calculated: ∫[e^(−2mx) θ^2(x)+2θ(x)θ(−x)+e^(−2mx)θ^2(−x)]dx from -∞ to ∞ where θ(x) is the unit step function with its amplitude 0 everywhere before x=0 and θ(−x) is the unit step function with its amplitude 0 everywhere after x=0In Introduction to Quantum Mechanics...
  11. B

    Laplace Transform for shifted Unit Step Function

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  12. I

    Laplace transform of unit step function

    Homework Statement The Attempt at a Solution I know that u(t) is a unit step function and holds a value of either 0 or 1. In laplace transform, when we integrate f(t) from 0 to infinity, we take u(t) to be 1. In this case, since u(t) is u(-t), does this mean it holds a value of 0? Does not...
  13. paulmdrdo1

    MHB Solve Unit Step Function Algebraically - Help Needed

    How would I go about algebraically (not graphically) performing operations on two or more Heaviside function.H(x) = {0, if x<0 ...{1, if x>=0 ex. define the function piecewise and graph. a.) (x+1)*H(x+1)-x*H(x) b.) (x+1)*H(x+1) i'm having a hard time solving this please help me!
  14. A

    Finding impedance that is unit step function

    Homework Statement Consider the following circuit which uses ideal components. Prior to t=0 switch S is open. Then suddenly at t=0 switch S is closed. Find the impedance Z_{2} such that the system output is a unit step function of voltage. Be certain to show all components used to construct...
  15. J

    FIR filters, impulse responses and unit step function

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  16. M

    Differential Equation with unit step function and Laplace transforms

    EDIT: Nevermind I see what I did wrong near the end. Homework Statement x'' + 4x = f(t) Where f(t) is 1 if t is between 0 and π, 0 if t > π. Initial conditions are x(0) = x'(0) = 0. Homework Equations Transform of a derivative: L(f^{(n)}(t)) = s^nF(s) - s^{n-1}f(0) -...-f^{n-1}(0)...
  17. lonewolf219

    How to find La Place transform of cos(x) * unit step function (x - pi)

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  18. J

    Can the DTFT of a Unit Step Function be Simplified for u(n) - u(n-L)?

    So I'm trying to find the DTFT of the following; where u(n) is the unit step function. u \left( n \right) =\cases{0&$n<0$\cr 1&$0\leq n$\cr} I want to find the DTFT of u \left( n \right) -2\,u \left( n-8 \right) +u \left( n-16 \right) Which ends up being a piecewise defined function...
  19. C

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  20. G

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  21. S

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  22. N

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  23. F

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    Homework Statement y'' + 4y' + 2y = u_pi(t) + u_2pi(t) y(0) = 0 and y'(0) = 0.Homework Equations the step function equation: u_c(t) = u(t-c) --> (laplace) --> e^-cs/sThe Attempt at a Solution i am having major probs with getting my head around step function probs wrt laplace transforms. What...
  24. N

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    Homework Statement Could someone please explain this to me? I have read several notes on it, but do not really follow the reasoning: The Attempt at a Solution When t = 0, -1/s*e^-st = -1/s because e^0 = 1. When t goes to infinity is the part I do not fully understand. Why...
  25. alexmahone

    MHB Write f(x) in terms of the unit step function u(x)

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  26. R

    The second shifting theorem and the unit step function

    Homework Statement I am trying to do some revision for an upcoming exam and one question I am trying to figure out is Use the second shifting theorem to find the Laplace transfrom of the following function: f(t) = t2, t < 4 t, t ≥ 4 Homework Equations The Attempt at a...
  27. C

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  28. E

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    Homework Statement L{2t u(t-1)} Homework Equations L{g(t) u(t-c)} = e^-cs L{g(t+c)} The Attempt at a Solution L{2t u(t-1)}=e^-s L{2(t+1)} L{2(t+1)}=2/s^2+2/s L{2t u(t-1)} = e^-s {2/s^2 + 2/s} i think the whole attempt is wrong , I'm getting confused in this type of...
  29. S

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    Homework Statement f(t) = {0, if t<4 and (t-3)^3 if t\geq4 The Attempt at a Solution I feel like its pretty basic but i can't get it down I have u(t-4)(t-4)^3 Can i change it to u(t-4)^4? Then do i multiply it out and take the laplace? If someone can work it out for me that would...
  30. icystrike

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  31. R

    Unit Step Function U(t): No Transient Signals

    unit step function u(t) does not contain any transient signal
  32. M

    Laplace transform unit step function

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  33. T

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    Homework Statement The Dirac function (unit impulse) is defined as \delta(t) = 0 where t \neq 0 the integration of d(t) between -ve inf and +ve inf is 1. Now I picture this as a rectangle with no width and infinite height. In fact I think of the width (along the x axis) as (1/inf =...
  34. R

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  35. J

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    Homework Statement I'm trying to take the laplace transfrom of t H(t) where H(t) is the unit step function. Also, in a separate problem I get e^{-t} H(t) - e^{-t}H(t-1) and I am wondering how to manipulate it properly Homework Equations L \{ f(t-a) H(t-a) \} = e^{as}F(s) The...
  36. A

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  37. D

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  38. K

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  39. V

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  40. C

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  41. U

    Trying to integrate a summation of a unit step function.

    Homework Statement Define I(x)= I( x - x_n ) = { 0 , when x < x_n { 1, when x >= x_n. Let f be the monotone function on [0,1] defined by f(x) = \sum_{n=1}^{\infty} \frac{1}{2^n} I ( x - x_n) where x_n = \frac {n}{n+1} , n \in \mathbb{N} . Find \int_0^1 f(x) dx ...
  42. S

    MATLAB Creating a unit step function in Matlab

    Hi Guys, I am trying to create a basic unit step function in Matlab that needs to be in the range of" -5 <= x <= 5 I need this to be done via a function and not piece together using different intervals and it needs to show the whole -5 to 5 interval. I am just beginning in Matlab and am...
  43. K

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  44. T

    Help with DE and UNIT STEP FUNCTION

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  45. M

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  46. J

    Unit Step Function Homework: Solve f(t) for t = -1

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  47. D

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  48. D

    Representing Unit Step Function in MATLAB

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  49. A

    What is the laplace transform of u(-t)?

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  50. T

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