Discrete-Time Signal Representations and Sequence Types

What a Discrete-Time Signal Is

A discrete-time signal is defined only at discrete instants. The independent variable takes a discrete set of values and is written $n$ (integer). The sample value is $x[n]$.

\[x[n] \text{ is defined for } n \in \mathbb{Z}, \quad \text{undefined for non-integer } n\]

A sequence is written with braces: ${x[n]}$. The braces are often dropped when there is no ambiguity.

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Four Representations

Every discrete-time signal can be written four ways. Take the sequence with values $x(-3)=-2$, $x(-2)=3$, $x(-1)=0$, $x(0)=-1$, $x(1)=2$, $x(2)=3$, $x(3)=1$.

1. Graphical

Stems at each integer $n$, height equal to the sample value. The $x$-axis is the sample index, the $y$-axis the amplitude.

2. Functional

\[x[n] = \begin{cases} -2, & n=-3 \\ 3, & n=-2 \\ 0, & n=-1 \\ -1, & n=0 \\ 2, & n=1 \\ 3, & n=2 \\ 1, & n=3 \\ 0, & \text{otherwise} \end{cases}\]

Or as a closed-form rule, e.g. $x[n]=n^2$ for $-3 \le n \le 4$.

3. Tabular

$n$	-3	-2	-1	0	1	2	3
$x[n]$	-2	3	0	-1	2	3	1

4. Sequence (vector)

\[x[n] = \{-2, 3, 0, \underset{\uparrow}{-1}, 2, 3, 1\}\]

The arrow marks the sample at $n=0$. If no arrow is shown, the first term is assumed to be at $n=0$. This convention matters: ${1,2,3}$ and ${1,2,\underset{\uparrow}{3}}$ are different signals.

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Real and Complex Sequences

A sequence is real if every $x[n]$ is real, otherwise complex.

A complex sequence splits into real and imaginary parts:

\[\{x[n]\} = \{x_{re}[n]\} + j\{x_{im}[n]\}\]

and its complex conjugate is ${x^*[n]} = {x_{re}[n]} - j{x_{im}[n]}$.

Example: ${y[n]}={e^{j0.3n}}={\cos 0.3n}+j{\sin 0.3n}$, so $y_{re}[n]=\cos 0.3n$ and $y_{im}[n]=\sin 0.3n$. Its conjugate is ${e^{-j0.3n}}$.

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Sampled-Data vs Digital Signals

Type	Time	Amplitude
Continuous-time	Continuous	Continuous
Sampled-data	Discrete	Continuous
Digital	Discrete	Discrete (quantized)

A practical DSP system works on digital signals: sampling makes time discrete, then quantization (rounding or truncation) makes amplitude discrete.

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Finite-Length and Infinite-Length Sequences

A finite-length (finite-duration) sequence is nonzero only for $N_1 \le n \le N_2$, with $-\infty < N_1 \le N_2 < \infty$. Its length is

\[N = N_2 - N_1 + 1\]

Example: $x[n]=n^2$ for $-3 \le n \le 4$ has length $4-(-3)+1=8$. In contrast $y[n]=\cos 0.4n$ is infinite-length.

A length-$N$ sequence is called an $N$-point sequence.

Zero-Padding

Appending zeros increases the length without changing the information:

\[x[n]=n^2,\ -3\le n\le 4 \quad \xrightarrow{\text{pad with 4 zeros}} \quad x_e[n]=\begin{cases} n^2, & -3\le n\le 4 \\ 0, & 5\le n\le 8 \end{cases}\]

The padded sequence has length $8+4=12$. Zero-padding is used everywhere in DFT/FFT work to avoid circular aliasing and to interpolate the spectrum.

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Right-Sided, Left-Sided, Causal, Anti-Causal

Type	Definition	Special case
Right-sided	$x[n]=0$ for $n<N_1$	If $N_1 \ge 0$ it is causal
Left-sided	$x[n]=0$ for $n>N_2$	If $N_2 \le 0$ it is anti-causal
Two-sided	Infinite in both directions	Neither
Finite-duration	Nonzero only on $N_1 \le n \le N_2$	Both right- and left-sided

Right-sided:  ... 0 0 0 | 1 2 3 4 ...     (nothing before N1)
Left-sided:   ... 4 3 2 1 | 0 0 0 ...     (nothing after N2)
Causal:       n = 0,1,2,... only
Anti-causal:  n = 0,-1,-2,... only

A causal sequence is exactly a sequence that a real-time system can produce: it never needs future input.

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Why These Classifications Matter

• Causality decides whether a filter is realizable in real time.
• Finite vs infinite length decides whether convolution is a finite sum or an infinite series.
• Right/left/two-sided decides the shape of the Z-transform ROC (exterior, interior, or annulus).
• Zero-padding decides whether DFT convolution equals linear convolution.

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Quick Check

For $x[n]={3, -1, \underset{\uparrow}{4}, 2}$:

1. Length? → 4
2. Causal? → No, it starts at $n=-1$
3. Right-sided? → Yes
4. $x[1]$? → 2
5. Padded to 6 points? → ${3,-1,4,2,0,0}$