# Is $\delta(n) = u(n) - u(n-1)$?

In my text book, Digital Signal Processing, Principles, Algorithms, and Applications by John Proakis, it asks the question to show how $\delta(n) = u(n) - u(n-1)$.

I can understand how this is true if "n" is an integer, but sometimes I see the step function as a continuous line instead of a fixed point at an instant. In that case, is it incorrect to think of $u(n)$ as something continuous?

• Does the text in question (who is the author, by the way?) use a very common (but by no means universal) notational convention: $x[n]$ means the $n$-th sample of a continuous-time signal $x(t)$, that is, $x[n] = x(nT)$ where $T$ is the time interval between samples? Or does the author routinely use $x(n)$ to mean either the $n$-th sample or the value of $x(t)$ at time $t = n$, because he, like Humpty Dumpty, scornfully believes that "When I use a word, it means just what I choose it to mean — neither more nor less." ?? – Dilip Sarwate Feb 25 '13 at 2:54
• @DilipSarwate That's what I'm confused about. It seems to switch back and forth. – Ci3 Feb 25 '13 at 5:00
• "... it seems to switch back and forth." Well, then, play Follow-my-leader and give the proof that makes sense, prefacing your answer by something like: "In this answer, I use $u$ and $\delta$ to mean sequences with $n$-th elements $u(n)$ and $\delta(n)$ where $n$ is an integer." Then write out your proof (probably just one or two lines) and end with a coda saying that the result to be proved is not correct if $u$ and $\delta$ are interpreted as continuous-time functions. That way you have C-ed YA against TAs eager to knock a few points off. – Dilip Sarwate Feb 25 '13 at 12:17
• The question only makes sense if it is referring to discrete sequences and, as Dilip said, the "n" terminology almost always implies discrete sequences, so I would assume that they mean discrete sequences. – Jim Clay Feb 25 '13 at 14:32