# What is the difference between y[n] and y(k) (using square brackets)?

Can I modify y[n] = αy[n-1] + x[n] the same way I would do it with y(k) --> Y(z)? I need H(z), but I don't know what is the difference between using (k), and (n). I often encountered [k] notation, what is the difference between (k) and [k]?

• Usually square brackets are used to indicate that the index is discrete, but conventions vary. – MBaz Jul 4 '19 at 17:34
• Ok, thanks. I'm thinking, maybe [n] would mean that the series is 0 when n<0, and some function f, f(n) when n>0? And h[k] could be positive or negative where ever. Would that make sense? – Immo Jul 4 '19 at 17:43
• Could be, but that's not an universal convention. – MBaz Jul 4 '19 at 17:45
• Your recursive relaton seems akin to first order exponential averaging low-pass filter, or exponentially weighted moving average (EWMA) see dsp.stackexchange.com/q/36415/15892 or first order exponential averaging low-pass filter, exponentially weighted moving average (EWMA) – Laurent Duval Jul 4 '19 at 19:12

The notation for discrete-time signals, $$x[n]$$, was first noticed by me in the original Oppenheim and Schaffer (1975), even though i didn't see that book until 1981.

So before, discrete-time signals in DSP used the same notation in mathematics used for sequences (infinite or not), with the subscript, $$x_n$$. In both notations, the argument or subscript play the same role and are only integers.

$$x[n] \triangleq x_n \qquad \qquad x\in\mathbb{C} \qquad n\in\mathbb{Z}$$

Usually the signal, $$x[n]$$, is real but it need not be. But the argument, $$n$$, must be an integer which makes this no different than a sequence, $$x_n$$.

With the DTFT and DFT this notation was common pre-O&S,

\begin{align} X_k &\triangleq \sum\limits_{n=0}^{N-1} x_n e^{-i 2 \pi \frac{nk}{N}} \\ \\ x_n &= \frac{1}{N} \sum\limits_{k=0}^{N-1} X_k e^{+i 2 \pi \frac{nk}{N}} \\ \end{align}

or they would use that awful $$W_N^{nk}$$ notation.

Now the problem with the "$$x_n$$" notation is that you would get too many subscripts to deal with, if you had a signal that was a multi-element vector. And there was nothing in the notation to denote which subscript was the "time-like" variable. With analog signals, $$v(t)$$, this wasn't a problem. If you had a cable with 8 wires (plus ground), you could label them "$$v_1(t)$$" to "$$v_8(t)$$" and looking at $$v_m(t)$$, there was no doubt that it meant the "$$m$$th element that is a function of time, $$t$$." But with "$$x_{m,n}$$" (as opposed to the O&S way "$$x_m[n]$$") there is nothing, other than order, differentiating discrete time from the "channel" number. Add another dimension (such as the 2-dim pixel radiance as a function of time) and you have a worser mess:

$$x_{r,c}[n] \qquad \text{vs.} \qquad x_{r,c,n}$$

$$r$$ is row number, $$c$$ is column number, and $$n$$ is discrete-time.

So with this notation, there is a parity between a vector of continuous-time (a.k.a. "analog") signals $$x_m(t)$$ and discrete-time (a.k.a. "digital") signals $$x_m[n]$$. So, likely, whatever relationship one might see between the various analog signals $$x_m(t)$$ in a multi-conductor cable, I might expect to see between the various elements of the digital vector signal $$x_m[n]$$ and sampling would be no different than with a single-dimensional signal

$$x_m[n] \triangleq x_m(t)\bigg|_{t=nT} \qquad \qquad T=\frac{1}{f_\mathrm{s}}$$

So this provides for a consistency of notation along with a way to differentiate arguments that must be integers whereas whatever is contained within parenths $$f\big( \cdot \big)$$ is considered to have no such restriction (could be a real or complex argument). That is, to me, helpful.

So, if it's a very old DSP paper (pre-1978, perhaps), the notation you would see would almost certainly be $$x_n$$ for discrete-time signals and the DFT would be $$X_k$$ for discrete-frequency. But this has evolved to

\begin{align} X[k] &\triangleq \sum\limits_{n=0}^{N-1} x[n] e^{-j 2 \pi \frac{nk}{N}} \\ \\ x[n] &= \frac{1}{N} \sum\limits_{k=0}^{N-1} X[k] e^{+j 2 \pi \frac{nk}{N}} \\ \end{align}

and EEs will use "$$j$$", instead of "$$i$$" for the imaginary unit.

Math folks might use this notation:

\begin{align} \hat{f}_k &\triangleq \frac{1}{\sqrt{n}} \sum\limits_{j=0}^{n-1} f_j \, e^{-i 2 \pi \frac{jk}{n}} \\ \\ f_j &= \frac{1}{\sqrt{n}} \sum\limits_{k=0}^{n-1} \hat{f}_k \, e^{+i 2 \pi \frac{jk}{n}} \\ \end{align}

• You should really put commas in your multiple subscripts to distinguish them from products. – Cedron Dawg Jul 7 '19 at 2:54
• let's just say that subscripts are not always identical to a discrete argument in square brackets. i don't mind seeing math in the argument of a continuous-time or discrete-time functions. i.e. $$x\big[\lfloor t/T \rfloor\big]$$ but i don't wanna see math done in the actual subscripts. they should simply be integers (or integer variables) and nothing else. – robert bristow-johnson Jul 7 '19 at 3:00
• So says the man who laid down $c_{N/2}$ just yesterday. Tee-hee. Anyway, something like $Y_{2n}$ is much clearer as $Y_{2,n}$ if that is what is really meant. Crufty or crafty, you decide. Oh, and I'm not going to get into a long discussion about this in comments, either. ;-) – Cedron Dawg Jul 7 '19 at 3:21
• aw, you're right. what can i say? – robert bristow-johnson Jul 7 '19 at 3:26
• "Thanks for the upvote"? – Cedron Dawg Jul 7 '19 at 3:40

Based on Electrical Engineering use of DSP notation,

• $$y(k)$$ indicates a continous argument function $$y(\cdot)$$ evaluated at an integer argument $$k$$ (letters $$i,j,k,l,m,n$$ are preferred for integer variables, unless otherwise explicitly stated)

• $$y[n]$$ indicates a discrete argument function, a sequence, $$y[\cdot]$$ evaluated at its integer index $$n$$. Mathematically, this can be considered as the $$n+1$$ st element of the sequence $$y[n]$$ (assuming $$n$$ starts from $$0$$), but that's not the case in DSP as integer $$n$$ ranges from $$-\infty$$ to $$\infty$$.

• @robertbristow-johnson hmm what's the common convention ? Do I know it too ? may be you should edit... – Fat32 Jul 6 '19 at 20:18
• what i meant by the common mathematical convention is $y_n$ instead of $y[k]$ or $y(n)$. – robert bristow-johnson Jul 7 '19 at 4:02
• @robertbristow-johnson your answer is more complete and historically relevant of course. I assumed no comprehensive treatment of mathematical subscript and superscript notation and its link to premature DSP; which is merely a convention. The question was about y(k) and y[n] and I just left it there... Still many authors (especially in Probability, Statistics, Information Theory, Coding, Communication etc), prefer $x_n$ instead of $x[n]$ to refer to the elements of a vector / sequence. As long as clear from the context (which should be) then there's no problem to use any convention. – Fat32 Jul 7 '19 at 10:55
• A small observation that helps me distinguish the usage case is that parentheses symbol ( ) is curved, and 'continuous' in shape, while brackets [ ] are sharp-edged, for 'discrete' function mapping. – Bryan W Jan 27 at 5:57
• @BryanW yes that's (possibly) the very reason of selecting $[~]$ for discrete time sequences... – Fat32 Jan 27 at 19:06