In this topic we will introduce sequences and discrete signals by

1. defining sequences and discrete signals,
2. describing mathematically defined sequences or discrete signals,
3. defining series,
4. describing common operations on sequences,
5. describing properties of sequences and discrete signals,
6. a summary, and
7. exercises.

1. Sequences and discrete signals

Suppose you have a temperature sensor, and you are periodically reading that sensor, for example, once per hour or once every 92 seconds. This will give you a sequence or discrete signal of temperature readings such as

$(21.2, 21.3, 21.2, 21.2, 21.3, 25.7, 36.9, \ldots)$.

There are two different interpretations of a sequences or signals:

1. When dealing with mathematical sequences, the first is the initial entry and subsequent entries are said to be subsequent to prior entries. This is often the interpretation when dealing with mathematically defined sequences such as the Fibonacci sequence $(1, 1, 2, 3, 5, 8, 13, \ldots)$. There is often a question of the behavior of the sequence as you continue along the entries.
2. When dealing with discrete signals, each subsequent entry is the next reading of, for example, a sensor. At some point in time, the sensor is turned on and will begin to periodically send information about a specific value, be it temperature, wind speed, direction, etc.

In all real situations, such discrete signals are always finite in length; for example, you may have collected all such temperatures between January 17th, 1985 to June 19, 2015; or all temperatures from 10 o'clock this morning when you turned the device on until the present. In such cases, a sequence or signal is said to be finite. However, when you are designing a system, you will not know what these readings will be, but you will still want to represent them. In that case, we can do so symbolically:

$t = (t[0], t[1], t[2], t[3], t[4], t[5], \ldots)$

In the sequence $t$, $t[0]$ is the first or initial reading, and each subsequent reading is the next reading of the sensor. The natural number ($0$, $1$, $2$, etc., see ISO 80000-2) is referred to as the index and the individual values in the sequence are called entries or terms.

If the sequence or signal is finite, we could represent it as

$t = (t[0], t[1], t[2], t[3], t[4], t[5], \ldots, t[n])$

In the design process, however, you may not know how long the readings will be taking: perhaps 10 minutes, perhaps 10 years. It would be really poor form to design, for example, a pacemaker that is only guaranteed to work exactly one year. Thus, in design, we may be concerned about infinite sequences or signals:

$t = (t[0], t[1], t[2], t[3], t[4], t[5], \ldots, )$

where $t[k]$ is assumed to have a value for each integer $k$ greater than or equal to zero. The natural number is referred to as the index and the $t[k]$ is the entry associated with the $k$^th index.

For the next two topics on arithmetic and geometric series, we will usually use the sequences

$x = (x[0], x[1], x[2], x[3], x[4], x[5], \ldots, )$
$y = (y[0], y[1], y[2], y[3], y[4], y[5], \ldots, )$

and we will specify whether or not the sequence is finite or infinite.

Thus, if $x$ is an infinite discrete signal, then $x[k]$ has a value for all integers $k$ greater than or equal to zero; while if $y$ is a finite discrete signal, then $y[k]$ only takes on values for integer values of $k$ going from $0$ to $n$.

2. Mathematically defined sequences or discrete signals

In the real world, there are no exact functions, no exact surfaces, but we can model the shape of the wing of an aircraft with polynomials, and we can model growth and decay with exponential functions.

Similarly, actual temperatures can never be predicted—they will vary, but we can never-the-less model sequences or discrete signals with mathematically defined signals. In the next two topics, we will look at two types of mathematically defined sequences or discrete signals:

1. arithmetic sequences, and
2. geometric sequences.

There are three means of mathematically defining sequences:

• explicitly defined,
• recursively defined, and
• defined based on another sequence.

We will look at both of these.

2.1 Explicitly defined sequences and discrete signals

A sequence $x$ may be written as an explicit formula based on the index; for example:

$x[k] = \frac{3.2}{k + 1}$ or
$y[k] = 2^{-k}$.

In such a case, it is reasonably simple to calculate the $k$^th entry. This is the preferable means of defining such sequences, as it allows you to calculate arbitrary entries and it is easier to describe the behavior of the series as the index becomes large.

The sequences $x$ and $y$ start with

$x = (3.2, 1.6, 1.0667, 0.8, 0.64, 0.5333, 0.4571, 0.4, 0.3556, 0.32, 0.2909, \ldots)$
$y = (1, 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.0078125, 0.00390625, 0.001953125, \ldots)$.

Mathematically defined sequences are often infinite by definition, but nothing stops us from terminating such a sequence.

2.2 Recursively defined sequences and discrete signals

In contrast with mathematically defined sequences, it is also possible to define future values based off of present values: for example, if $x[0]$ is your bank account in 2016, and your money is sitting in a GIC yielding 1.5% interest, then $x[1] = 0.015x[0]$, and in general (assuming you don't change accounts or banks), $x[k] = 0.015x[k - 1]$.

To define such a sequence, you must however know how much money you had in the bank at some initial time: if you don't know how much money you had in the bank last year, you can't determine how much you will have this year.

Such a definition is said to be recursive.

The word recursive is based off the Latin verb recurrence meaning "to run back".

Likely the most famous recursively defined sequence is the Fibonacci sequence $f$:

$f[0] = 1$ and $f[1] = 1$ and $f[k] = f[k - 1] + f[k - 2]$.

This yields the sequence starting with

$f = (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \ldots)$.

In some cases, it is desirable to determine whether or not a recursively defined sequence can be converted to an explicitly defined sequence. For example, the Fibonacci sequence can also be defined as

$f[k] = \frac{1}{5}(\phi + 2)\phi^k + \frac{1}{5}(3 - \phi)(1 - \phi)^k$

where $\phi = \frac{\sqrt{5} - 1}{2}$ is the golden ratio.

2.3 Sequences based on other sequences

Given a sequence $x$, we may define a new sequence $y$ based on the entries of $x$. The easiest is when the entries of $y$ are defined as functions of the entries in $x$:

$y[k] = f(x[k])$

for some function $x$. For example, if you wanted to amplify a discrete signal, you may simply define $y$ as being $10$ times the entries of $x$:

$y[k] = 10\cdot x[k]$.

Alternatively, suppose that $t$ was a record of the number of milliseconds since a device was turned on:

$t = (0, 1, 2, 3, 4, 5, \cdots)$.

In this case, you could define a slowly changing sinusoid as follows:

$y[k] = \sin(0.00052 \cdot t[k])$.

Suppose, however, you wanted to define $y$ to be the sum of the entries of $x$. In this case, you could create a recursive definition:

$y[k] = y[k - 1] + x[k]$.

For example, if $x$ was given as:

$x = (0.838, 0.117, 0.346, 0.619, 0.334, 0.593, 0.288, 0.346, 0.252, 0.169, 0.138, \cdots)$,

then $y$ would be equal to the sequence

$y = (0.838, 0.955, 1.3010, 1.920, 2.254, 2.847, 3.135, 3.481, 3.733, 3.902, 4.040, \cdots)$.

Another example of a sequence is to define a sequence that gives you the difference between the current value of $x[k]$ and the previous value $x[k - 1]$:

$d[0] = 0$ and $d[k] = x[k] - x[k - 1]$ for $k = 1, 2, 3, \ldots$.

In this case, if we were trying to implement this on a computer, we must remember, if nothing else, the second most recent entry, at least until we use it. For the above sequence,

$d = (0, -0.721, 0.229, 0.273, -0.285, 0.259, -0.305, 0.058, -0.094, -0.083, -0.031$, \ldots).

Now, what this tells you is the instantaneous rate of change. As this last sequence was random, the differences vary wildly, but suppose we had a sequence of temperatures once an hour on a warm summer day:

$t = (21.300, 21.744, 22.533, 23.507, 24.453, 25.160, 25.462, 25.283, 24.651, 23.700, 22.631, \ldots)$,

in which case

$d = (0, 0.444, 0.789, 0.974, 0.946, 0.707, 0.302, -0.179, -0.632, -0.951, -1.069, \ldots)$,

so initially we observe the temperature increasing, but then it starts to cool off.

Another question is whether or not the data is concave up or concave down. This can be determined by the sequence defined as:

$c[0] = c[1] = 0$ and $c[k] = x[k] - 2x[k - 1] + x[k - 2]$ for $k = 2, 3, 4, \ldots$,

so for our above temperatures:

$c = (0, 0, 0.345, 0.185, -0.028, -0.239, -0.405, -0.481, -0.453, -0.319, -0.118, \ldots)$,

In your course on numerical analysis, you will see why this approximates the concavity, either positive (concave up) or negative (concave down).

3. Series

In many cases, it is necessary to add the values in a sequence or discrete signal: thus, given a finite sequence or finite discrete signal $x$, the series associated with $x$ is

$x[0] + x[1] + x[2] + x[3] + x[4] + \cdots + x[n]$.

If the sequence or discrete signal $y$ is infinite, we could write the series as

$y[0] + y[1] + y[2] + y[3] + y[4] + \cdots$.

This becomes tedious to print, so we often represent such series using summation notation:

$$x[0] + x[1] + x[2] + x[3] + \cdots + x[n] = \sum_{k = 0}^n x[k]$$

and

$$y[0] + y[1] + y[2] + y[3] + \cdots = \sum_{k = 0}^\infty y[k]$$

The lightning-bolt like character is the Greek capital letter sigma, and this denotes that this is a sum of the values $x[k]$ and $y[k]$ where in the first case, we add the values from $k = 0$ up to $k = n$; and in the second case, we add the values for all integers starting at $k = 0$.

Now, given a mathematically-defined sequence, it may be possible to determine both finite and infinite series:

$$\sum_{k = 0}^\infty 2^{-k} = 2$$ $$\sum_{k = 0}^\infty \frac{1}{(k + 1)^2} = \frac{pi^2}{6}$$ $$\sum_{k = 0}^n 1 = n + 1$$

Sometimes, the range of the sum is written to the right of the Greek sigma: $\sum_{k = 0}^n 2^k = 2^{n + 1} - 1$.

4. Operations on sequences (beyond secondary school)

We will quickly describe three operations on sequences or discrete signals:

1. multiplying a sequence by a real number (a scalar),
2. adding two sequences, and
3. delaying a signal.

4.1 Multiplying a sequence by a real number (a scalar)

Given a sequence or discrete signal $x$, we can multiply every entry in that sequence by a real number (or scalar) $r$. We represent this multiplied sequence by $rx$. Thus, if we started with the sequence

$x = (1, 0.5, 0.25, 0.125, 0.0625, \ldots)$,

then multiplying this sequence by $r$ yields the sequence

$rx = (r, 0.5r, 0.25r, 0.125r, 0.0625r, \ldots)$.

If $r > 1$, this makes every entry in the signal larger, and thus is a process we call amplification; while if $0 < r < 1$, this reduces the signal, and is a process called attenuation.

4.2 Adding two sequences

Given two sequences or discrete signals $x$ and $y$ with the same number of entries, we can add their corresponding entries to produce a new sequence. If both $x$ and $y$ have $n + 1$ entries, then

$x + y = (x[0] + y[0], x[1] + y[1], x[2] + y[2], \ldots, x[n] + y[n])$

while if both are infinite, their sum would be

$x + y = (x[0] + y[0], x[1] + y[1], x[2] + y[2], x[3] + y[3], \ldots)$.

4.3 The delay operator

The final operation we will look at is the delay operator. If $x$ is a sequence or discrete signal, $Dx$ is the original sequence with the first entry removed. If $x$ is a finite sequence, then $Dx$ is a shorter sequence, but if $x$ is infinite, $Dx$ remains infinite:

For example, if $x = (3.2, 3.9, 3.7, 4.1, 4.2, 3.8)$, then $Dx = (3.9, 3.7, 4.1, 4.2, 3.8)$. If, however, if the signal is infinite, a delayed infinite signal is still infinite. For example, if

$x = (x[0], x[1], x[2], x[3], x[4], \ldots)$,

then

$Dx = (x[1], x[2], x[3], x[4], x[5], \ldots)$.

If an infinite sequence $x$ is explicitly defined, then $x[k] = f(k)$ for some function $f$. In that case, the $k$ entry of $Dx$ is $(Dx)[k] = f(k + 1)$. For example, if $x[k] = 2^{-k}$, then the $k$^th entry of $Dx$ is $(Dx)[k] = 2^{-(k + 1)} = 2^{-k - 1}$.

5. Properties of sequences and discrete signals

If a mathematically defined infinite sequence has entries that grow to infinity in absolute value, then we say the sequence is unbounded; for example, all of the sequences

• $(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, \ldots)$
• $(0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, \ldots)$
• $(1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, \ldots)$
• $(-1, 3, -27, 81, -254, 729, -2187, 6561, -19683, 59049, \ldots)$

are unbounded (of course, under the assumption that the pattern is followed).

If, however, a mathematically defined infinite sequence does not grow to infinity in absolute value, then we say the sequence is bounded; for example, all of the sequences

• $(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \ldots)$
• $(2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, \ldots)$
• $(20, 2, 0.2, 0.02, 0.002, 0.0002, 0.00002, 0.000002, \ldots)$
• $(-3900, 390, -39, 3.9, -0.39, 0.039, -0.0039, 0.00039, -0.000039, \ldots)$
• $(1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, \ldots)$

are bounded.

Temperatures tend to be bounded, while the distance a car has travelled since it comes off of the assembly line (or, if you prefer, the nautical miles an aircraft carrier has sailed or kilometers an A380 has flown) is theoretically bounded.

6. Summary

The use of infinite sequences and series is essential to modelling discrete systems in the real world. We will now go on to look at both arithmetic and geometric sequences and series.

7. Exercises

1. To be completed.