# Unit impulse added to non-zero interval of a function - graphical representation?

Followup to this question, let the ramp be $$r_0(t)$$. We seek to plot $$x(t) = r_0(t) + \delta(t + 1)$$. Should it be as in left, right, or neither? Is there a convention?

Note, in either case we could use either arrow size or label it to show its value; this question's about the tail's placement. Others validly pointed that the label works best in general for $$(a + jb)\delta(t)$$.

• I've never seen A, and to me it's also unclear what the area of the Dirac impulse should be in that representation. A doesn't make as much sense to me as B, because the function and the Dirac impulse don't add up as values. Also, I've seen B before, would need to look up where ... Commented May 16, 2023 at 18:45
• @MattL. You're not proposing B? Are you proposing putting a hole in $r_0(t)$? Because I'd certainly take an issue with that; it destroys information in $r_0(t)$, which could be any other value - maybe not an issue for $r_0$, but for $1 / (t + 1)$. As for A, the area would equal the height rather than its y-value. As for "don't add the values", the integrals still add, so any operational result upon $x(t)$ is additive. Commented May 16, 2023 at 18:50
• @MattL. I somehow read "I've never seen B before", nevermind. Yes a ref would be good. Commented May 16, 2023 at 20:45

I favor $$A$$ because:

1. It preserves one-to-one graphical property of mappings, as in one input graphically corresponds to one output. That I know of, there aren't any other function-like mathematical objects that require different treatment.
2. It preserves graphical properties of coordinate transformations: $$t \rightarrow t + t_0$$ stays a horizontal shift, $$x(t) \rightarrow x(t) + x_0$$ stays a vertical shift. Off-topic, but if height denotes value, then $$x(t) \rightarrow ax(t)$$ stays a vertical stretch, and $$x(t) \rightarrow x(at)$$ stays a horizontal stretch (... as in, $$a \cdot 0 = 0$$). $$B$$ does all of these except vertical shift.
3. $$0$$ isn't special in any value sense, so planting the tail there, immovably, shouldn't be special either.
4. Representative power: let $$x(t) = 1000 + \sin(t) + \delta(t)$$. Then we can't draw without either omitting $$\delta(t)$$ or making the plot look like $$1000 + \delta(t)$$.

I like $$B$$ because it's cleaner to keep the deltas separate from the "actual" function and keep them visually aligned, also in its emphasis of the non-functional nature of $$\delta(t)$$. Point 4 can be resolved by shifting the reference to some other visually convenient $$y=y_0$$. Others are free to repeat this information.

What is universally accepted is that the value shown for a continuous time impulse represents the area within it and not its actual height (its actual height is infinite, and the area is the integration over that zero range). This is the case for B but that isn't properly represented by A, as the area added by the continuous function over this same range is 0, so the represented impulse should not have a larger value. That alone makes A incorrect: as drawn without any other indication of the impulses scale (area), it represents an impulse with an area of 2.5, but we know from the math given that the area of that particular impulse is still 1.

That said what is typical is for the unit impulse to be drawn with a value next to it indicating its area (which can and often is a complex quantity). This wouldn’t add in area to any continuous function; so whatever convention is used should appear the same as if we drew the impulse with or without the other function added.

• I'm not well-versed on impulse conventions by any means, but I've not heard of this "universal acceptance" before. Have a reference? Also if we follow the label as opposed to height convention, there doesn't appear to be an issue. Commented May 17, 2023 at 15:22
• lpsa.swarthmore.edu/BackGround/ImpulseFunc/ImpFunc.html. If not height then we need to add a number right next to the impulse to denote its area (the area is relevant as representing scaled impulses in general, such as being able to distinguish $\delta(t)$ from $5\delta(t)$ for example). But to show that it adds to a continuous function is quite misleading, since it is the area we are dealing with and that hasn't increased. Commented May 17, 2023 at 15:24
• That's the standard representation in which "arrow height" and "y value" coincide, but the reference doesn't distinguish. What would qualify is an explicit discussion. Concerning misleading, if it's established that the size of the arrow or its label denotes its value, there shouldn't be an issue. In absence of clarification, it's a fair sentiment, as are the advantages of A over B I discuss. Commented May 17, 2023 at 15:30
• Yes in neither of your two options shown you don’t indicate the area so imply that it is as aligned with the vertical axis. What isn’t show is the option with the value right next to the impulse- that is what I would typically see (I added a sentence making that clearer). Commented May 17, 2023 at 15:52
• I mainly opened this per discussion with MattL where I acknowledged the labeling, I also wasn't aware of the "y value" convention so I figured it redundant to mention. Fair, updated. Commented May 17, 2023 at 16:01

I think both of your examples loses the flavor of a unit impulse, in that it has infinite amplitude.

I'd make it an arrow, higher than any amplitude of the continuous function in the graph, with a number next to it to indicate its amplitude.

I'd also make a point to thoroughly describe my function with text and with equations (i.e., $$x = \left (- \frac 3 2 x \right) \bmod 2 + \delta (t + 1)$$) to give my audience the most opportunities to decipher what I'm trying to tell them. Then I'd proof-read the heck out of it to make sure I'm not introducing any real or apparent contradictions between graph, text, and equations.

This is really a problem of communicating complicated concepts to humans rather than DSP math per se., so there is no one right way to do it -- following convention is nice, but at the end of the day if you rigidly follow convention and your audience is confused, you've done them a disservice.