The

**unit ramp function**t(t), is a ramp function with a constant slope of 1. Widely used in signal processing, the function forms a building block for more complex signals.

## Expression as an Integral

The unit ramp function is the integral of the unit step function μ(t), so can be expressed as the following integral:

The unit ramp function can also be obtained by integrating the unit impulse function twice.

## Values of the Unit Ramp Function

The function can be expressed mathematically as:

Or, alternatively, by angles. The unit ramp is horizontal with one shift (in an anticlockwise direction) at t = 0 where the function takes on a 45 degree angle to infinity (Singh et. al, 2013).

## Shifted Unit Ramp Function

The unit ramp function usually starts at zero. However, it can also shift along the x-axis (in the positive direction). This function is called a **delayed ramp function**, because of the delay in the start time, at t = a. The function’s values will obviously not be zero, as in the above definition. Instead, the shifted unit ramp is defined as (Bakshi, 2009):

- f(t) = (t – a), for t ≥ a
- f(t) = 0, for t < a.

## References

Bakshi, U. (2009). Circuit Theory. Technical Publications Pune.

Singh, S. (2013). Proof of Spâ€™s… Ramp Function with the Help of Examples. International Journal of Engineering and Innovative Technology (IJEIT) Volume 2, Issue 7, January 2013