Unit impulse and unit step
The impulse is not a function with a shape: it is the limit of any family of pulses that narrows while keeping area one. All you need to know about it is how it behaves inside an integral.
01The step, which holds no mystery
Let us start with the easy one. The unit step is zero before and one after:
Its main use is to cut off signals. Multiplying by \( u(t) \) turns everything earlier into zero, and subtracting two shifted steps builds a pulse: \( \operatorname{rect} = u(t) - u(t - T) \). It is the standard way to write piecewise-defined signals on a single line.
02The impulse, which does
In discrete time it is simple too: \( \delta[n] \) equals 1 at \( n = 0 \) and zero elsewhere. A signal like any other.
In continuous time, on the other hand, \( \delta(t) \) is not a function. There is no function that is zero everywhere except at one point and has integral one: changing the value at a single point does not change any integral. What does exist is a limit, and more precisely a distribution: an object that is defined by how it acts inside an integral.
Any family of pulses that narrow while keeping area one gets to the same place: a rectangle, a triangle, a Gaussian, a sinc. The shape does not matter. What matters is that the area stays at one and the width tends to zero.
A pulse of area one that keeps narrowing. Above, the pulse next to a smooth signal; below, the result of \( \int x(t)\,p_\Delta(t - t_0)\,dt \), which has to approach \( x(t_0) \) regardless of the shape of the pulse.
03The properties that get used
| Property | Expression | What it is for |
|---|---|---|
| Sifting | \( \int x(t)\delta(t-t_0)dt = x(t_0) \) | Extracting a point value |
| Product | \( x(t)\,\delta(t-t_0) = x(t_0)\,\delta(t-t_0) \) | Simplifying before integrating |
| Area | \( \int \delta(t)\,dt = 1 \) | It is what defines it |
| Scaling | \( \delta(at) = \dfrac{1}{|a|}\,\delta(t) \) | Changes of variable |
| Parity | \( \delta(-t) = \delta(t) \) | It is even |
| Convolution | \( x(t) * \delta(t - t_0) = x(t - t_0) \) | Convolving with an impulse shifts |
The scaling one is surprising: narrowing the impulse makes it taller, because the area has to keep equaling one. And the last row is the one that explains why \( \delta(t) \) is the identity element of convolution: convolving with it changes nothing, and with a shifted version, it only shifts.
Since \( \int \delta(t)dt = 1 \) and \( dt \) is measured in seconds, \( \delta(t) \) has units of 1/second. It is not dimensionless. That is why the impulse response of a circuit is measured in “volts per volt-second”, and why in practice it is not excited with an impulse but with a step: a pulse of area one and 1 μs width would be a million volts tall.
04One is the derivative of the other
The relationship between the two is the one that organizes everything:
The derivative of a step is zero everywhere except at the jump, where it is infinite: that is exactly an impulse. And in discrete time the relationship is the same with differences and sums:
From this follows something very practical. Since an LTI system commutes with the derivative, the impulse response is the derivative of the step response. And that solves a real lab problem: an impulse cannot be generated, but a step is generated with a switch. The step response is measured and then differentiated.
It depends on the system, not on the pulse. What is required is that the pulse be much narrower than the fastest time constant of the system: as a rule, one tenth. For a circuit with \( \tau = 1 \) ms, a 100 μs pulse is enough, and the system cannot tell that input from a true impulse. That is why in practice the impulse is approximated without drama, as long as it is known beforehand how fast the system is.
05In the lab
Compute \( \int_{-\infty}^{\infty} (t^2 + 3)\,\delta(t - 2)\,dt \), \( \int_0^{5} e^{-t}\delta(t-3)dt \) and \( \int_0^{5} e^{-t}\delta(t-7)dt \). Explain why the last one gives zero.
Measure the step response of an RC with the oscilloscope, export the data and differentiate it numerically. Compare the result with the theoretical expression \( h(t) = \frac{1}{RC}e^{-t/RC}u(t) \).
Excite the same RC with pulses of equal area and widths of \( \tau \), \( \tau/10 \) and \( \tau/100 \). Check that from \( \tau/10 \) on the response no longer changes shape: the system cannot tell those pulses from an impulse.
06Common mistakes
- Treating \( \delta(t) \) as a function. It only makes sense inside an integral. Its “value” at zero is not a number.
- Forgetting the \( 1/|a| \) in \( \delta(at) \). Narrowing raises the height.
- Sifting with an impulse whose argument falls outside the interval. If \( t_0 \) is not in the range, the result is zero.
- Believing that \( \delta[n] \) is also a distribution. In discrete time it is an ordinary, everyday signal.
- Thinking the impulse is dimensionless. It has units of 1/second.
07Self-assessment
What is \( \int (t^2+3)\,\delta(t-2)\,dt \)?
7: it is the value of \( t^2 + 3 \) evaluated at \( t = 2 \).
Why is \( \delta(t) \) not a function?
Because no function that is zero except at one point can have integral one. It is a distribution, defined by how it acts inside an integral.
What is \( \delta(3t) \)?
\( \frac{1}{3}\delta(t) \). The factor \( 1/|a| \) preserves the area.
What is the relationship between \( h(t) \) and the step response?
\( h(t) \) is its derivative. That is why in the lab the step is measured and then differentiated.
What does \( x(t) * \delta(t - 3) \) give?
\( x(t - 3) \): convolving with a shifted impulse shifts the signal.
How wide can a pulse be and still pass for an impulse?
Much less than the fastest time constant of the system; as a rule, one tenth.
08Further reading
- Alan V. Oppenheim and Alan S. Willsky. Signals and Systems. 2nd ed. (Spanish edition, “Señales y sistemas”), Prentice Hall, 1998. The impulse defined as a limit of pulses and its properties, with the care the subject deserves.
- Ronald N. Bracewell. The Fourier Transform and Its Applications. 3rd ed., McGraw-Hill, 1999. The best treatment of the impulse and of distributions for engineering ever written.
- Simon Haykin and Barry Van Veen. Signals and Systems. (Spanish edition, “Señales y sistemas”), Limusa Wiley, 2001. The relationship between impulse response and step response, with measurement examples.