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

Question

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]?

Answer 1

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:

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

Answer 2

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 $\infty$.