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

    Derivative of unit step function

    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

    Laplace Transforms Involving: Unit-Step, and Ramp Functions

    Homework Statement Here is an imgur link to my assignment: http://imgur.com/N0l2Buk I also uploaded it as a picture and attached it to this post. Homework Equations u_c (t) = \begin{cases} 1 & \text{if } t \geq c \\ 0 & \text{if } t < c \end{cases} The Attempt at a Solution Question 1.1 -...
  7. J

    Laplace Transform question with unit step function

    Homework Statement Homework Equations Laplace Trasformations The Attempt at a Solution a. done b. f(t)= t -3*t*u(t-1) + 4*u(t-1) -3*u(t-2) -2*t*(t-2) c. 1/(s^2) - (3e^-s -2e^-2s)/(s^3) + (4e^-s -3e^-2s)/s d. 1/(s-1) * (1/(s^2) - (3e^-s -2e^-2s)/(s^3) + (4e^-s -3e^-2s)/s) These are the...
  8. sinaphysics

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

    Consider: u(t)=\begin{cases} 1\quad \quad \quad \quad t>0 \\ 0\quad \quad \quad \quad t<0 \end{cases} Now I want to calculate this: \int _{ 0 }^{ a }{ \frac { u(t)-u(t-a) }{ { t }^{ 2 } } } dt whereas: a>0 What is confusing me is this point that should our answer for the integral include...
  9. T

    Integral of Unit step function

    This isn't really so much of a specific example from a textbook, I just need to understand how to do this kind of integral and from that I can infer how to do all of the other problems. 1. Homework Statement The current problem I'm having trouble with is: Integral of (5u(t-1)) from T to -T...
  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

    Hello, I have a relatively simple question. after being unable to find it through google I have decided to ask you guys if you know what the Laplace transform of a unit step function that looks like this would look like Us(t-2) From tables, the Laplace transform for a regular units step...
  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

    Hey everyone, the question I am faced with is this: Which of the following expressions involving δ[n] is incorrect? where "m" is a non zero integer and u[n] is the unit step function. A. u[n-m] = δ[n] + u[n-m+1] B. x[n]δ[n-m] = x[n-m] C. δ[n] = u[n] - u[n-1] D. δ[n]δ[n-m] = 0...
  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)

    Homework Statement Find the La Place transform of cos(x)*(u(x-\pi)) Homework Equations L{u(t-a)}(s)=(e^(-as))/s The Attempt at a Solution I don't think I can just multiply this by the La Place transform of cos (x), which is s/(s^2) ?
  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

    Derivative of a unit step function

    So I'm busy struggling with some worked examples in my signals class. I understand the theory from the notes and textbook but I cannot seem to apply them to proper examples. We are asked to find the derivative of g(t) = (1-e^(-t))*u(t) where u(t) is a unit step function. I know the...
  20. G

    Is my unit step function correct for solving this differential equation?

    Homework Statement y''-4y'-32y={1 when 0<=t<1 and 0 when 1<=t y(0)=y'(0)=0 Homework Equations The Attempt at a Solution s2L(y) -4sL(y)-32 L(y)=u1(t) I am just struggling to figure out if my unit step function is correct. Solving for L(y) I get: (e-s) / (s(s2 -4s-32))...
  21. S

    Inverse Laplace transform with unit step function

    Hello again. First off, I wasn't sure how to say this in the title but I'm not taking the inverse Laplace transform of a unit step function. I'm taking the Laplace transform of something that comes out to the unit step function. I have this question, which is a similar version of the...
  22. N

    Problem with unit step function

    Homework Statement I'm having trouble with part b and part d, where there is some kind of ramp function involved http://img845.imageshack.us/img845/7507/76500775.jpg The Attempt at a Solution For part b, I calculated the gradient of that ramp, and the intercept which gives y = -x...
  23. F

    Confused about unit step function wrt laplace transforms

    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

    Laplace transform of unit step function

    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)

    Write f(x) in terms of the unit step function u(x). $u(x)=\left\{ \begin{array}{rcl} 1\ &\text{if}& \ x\geq 0 \\ 0\ &\text{if}& \ x<0\end{array} \right.$ $f(x)=\left\{ \begin{array}{rcl} 1\ &\text{if}& \ 2n\le x\le 2n+1 \\ 0\ &\text{elsewhere}\end{array} \right.$
  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

    Unit step function from piecewise

    Homework Statement f(x) = \begin{cases} 0, & t < \pi \\ t - \pi , & \pi \leq t < 2 \pi \\ 0, & t \geq 2 \pi \end{cases} Homework Equations Unit step function: u_c(t) = \begin{cases} 0, & t < c \\ 1 , & t \geq c \\ \end{cases} The Attempt at a Solution u_{\pi}(t)(t-\pi) - u_{2...
  28. E

    Problem with Laplace transform unit step function questions

    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

    Laplace of a Unit Step Function

    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

    Solving a Differential Equation with Unit Step Function

    Homework Statement Its not homework anyway: I'm asked to find the solution to the differential equation: i'' + 2i = f(t) i'(0)=i(0)=0 f(t) = u(t-10) - u(t-20) Unit step function (I've found in part a of the question) Then I've gotten: \mathscr{L}(i) =...
  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

    What is the laplace transform of a function that is 1 from 0 to 10 and 0 elsewhere? I know that this can be represented by the step function U(10-t)U(t)...but how do i find the laplace transform of this?
  33. T

    Signals & Comms: Unit step function - dirac

    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

    Integral form of the unit step function

    Homework Statement I need to show that the unit step function (\Theta(s) = 0 for s<0, 1 for s>0) can be written as \Theta(s)=\frac{1}{2\pi i} \int_{-\infty}^{\infty} dx \frac{e^{ixs}}{x-i0}. Homework Equations - The Attempt at a Solution Firstly, I'm unsure about what "x-i0" actually...
  35. J

    Laplace transform with unit step function

    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

    Unit Step Function: u(-t) Meaning & Negative Time

    what does the u(-t) mean in unit step funtion in basic function of signal and system?how can be time taken as -ve?
  37. D

    Laplace Transform of unit step function HELP

    Laplace Transform of unit step function HELP! Homework Statement f(t)= e^t on 0<=t<1 . . . . t on 1<=t<2 . . . . sin(t) on 2<=t<infinity Homework Equations Unit Step Function The Attempt at a Solution Here is my attempt at a solution...
  38. K

    Integration with a unit step function

    Hi, i have a problem with integration a function with a unit step function. Homework Statement Given, Refer to the image, i dun understand is that u(t) is equal to 1 from a definite integration from -\infty to \infty since u(t)=1 from -\infty to 0 and u(t)=0 from 0 to \infty...
  39. V

    Is the Resultant of Combining Unit Step Functions Still a Unit Step Function?

    if you subtract two delayed unit step functions, is the resultant a unit step function too? what is the value at the last point? 0 or 1? similarly, if you add 2 unit step functions will the magnitude of the resultant funtion be 1 or 2?
  40. C

    Unit Step Function and Matlab

    Homework Statement plot this function: cos(pi*n)*u(n) The Attempt at a Solution n=0:1:10; y=(n>=0); x(n) = cos(pi()*n)*y figure(1) plot(y)
  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

    Can the unit step function express e^t between 0 and 2?

    Homework Statement Express f(t) = e^t, 0<t<2, using the unit step function 2. The attempt at a solution e^t*u(t-2) is an expression for a graph of the function that is zero until t=2. My guess is e^t*u(t+2)
  44. T

    Help with DE and UNIT STEP FUNCTION

    I'm not sure how to solve a differential equation with unit step function, for example: x'' + 2x' + x = 10t*u(t), where x(0)=1 and x'(0)=0 Do I just ignore the u(t) and solve it regularly by normal integration?
  45. M

    Heaviside (Unit step function) question

    Homework Statement The function f(t) is defined for t>=0 by f(t) = 1 for 0<= t <= 1 , t-2 for 1 <=t <= 2 and 0 for t >2 Express f(t) in terms of the Heaviside function and hence or otherwise find L(f(t)), the Laplace transform of f(t) Homework Equations The Attempt at a...
  46. J

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

    Homework Statement I'm having a bit of difficulty understanding the unit step function For example f(t) = 6u(-t) + 6u(t+1) - 3u(t+2) t = -1 Homework Equations The Attempt at a Solution If t = -1 then the way I would do the problem is say that if u(t) returns a negative number...
  47. D

    Sketch Signal: (t-4)[u(t-2)-u(t-4)]

    The question says to sketch the signal (t-4)[u(t-2)-u(t-4)]. I know that the inner part is a delay of two and a delay of four, but I don't know what to do with the (t-4)...does it have anything to do with the slope?
  48. D

    Representing Unit Step Function in MATLAB

    How do you represent a unit step in MATLAB as ONE function, y , for example. I know only how to graph multiple vectors on top of the same graph (using hold on/off) and getting the graph output, but i can't represent it as a function itself. Any ideas?
  49. A

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

    Homework Statement Find the laplace transform of u(-t) Homework Equations The Attempt at a Solution For u(t), the laplace transform of it is 1/s, basically taking the integral of e^-st from 0 to infinity. In this case, since the unit step function approaches from the negative...
  50. T

    Is a Unit Step Function or Series the Solution to this Equation?

    Homework Statement I have an equation that has the following values at different intervals: It is: r when 0<x<2Pi r - (1)d when 2Pi<x<4Pi r - (2)d when 4Pi<x<6Pi And so on. I want to find a function that encompasses this whole function. Unit functions / discontinuity functions are fine; as...