Discrete-Time Signals

Introduction

A discrete-time signal is a function defined only at integer values of the independent variable — typically denoted $x[n]$ where $n$ is an integer representing the sample index. Unlike continuous-time signals that have values at every instant, discrete-time signals exist only at specific, equally-spaced moments. This distinction is not merely academic — it is the foundation of all digital signal processing, since computers can only store and process numbers at discrete points.

Every digital audio file, every digital photograph, every piece of data processed by a computer is a discrete-time signal. Understanding how to represent, analyze, and manipulate these sequences is essential for modern engineering. For B.Tech students, this topic bridges the gap between the continuous-time theory (Laplace transforms, analog filters) and practical digital implementation.

Representation of Discrete-Time Signals

Functional Notation

A discrete-time signal is written as $x[n]$ where square brackets indicate discrete-time (as opposed to parentheses for continuous-time). The value $x[n]$ represents the amplitude of the signal at sample index $n$.

Sequence Notation

A signal can be listed as a sequence of values:

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

The arrow marks the $n = 0$ position. For example: $x[n] = \{2, \underset{\uparrow}{1}, 3, 4, 0\}$ means $x[-1]=2, x[0]=1, x[1]=3, x[2]=4, x[3]=0$.

Graphical Representation (Stem Plot)

Discrete signals are plotted as vertical stems at each integer $n$, with height equal to $x[n]$. This distinguishes them from continuous signals (smooth curves).

Fundamental Discrete Sequences

Unit Impulse (Kronecker Delta)

$$\delta[n] = \begin{cases} 1 & n = 0 \\ 0 & n \neq 0 \end{cases}$$

Unit Step

$$u[n] = \begin{cases} 1 & n \geq 0 \\ 0 & n < 0 \end{cases}$$

Unit Ramp

$$r[n] = n \cdot u[n] = \begin{cases} n & n \geq 0 \\ 0 & n < 0 \end{cases}$$

Exponential Sequence

$$x[n] = a^n u[n]$$

where $|a| < 1$ gives a decaying sequence and $|a| > 1$ gives a growing sequence.

Sinusoidal Sequence

$$x[n] = A\cos(\omega_0 n + \phi)$$

where $\omega_0$ is the digital frequency in radians per sample.

Any Signal as Sum of Shifted Impulses

The most important representation: any discrete signal can be written as a sum of shifted, scaled impulses:

$$x[n] = \sum_{k=-\infty}^{\infty} x[k] \cdot \delta[n-k]$$

This decomposition is the foundation of convolution — if we know the system's response to $\delta[n]$ (the impulse response $h[n]$), we can find the response to any input by superposition.

Classification of Discrete-Time Signals

Finite vs Infinite Length

• Finite-length: $x[n] \neq 0$ only for $N_1 \leq n \leq N_2$ (e.g., a windowed segment)
• Right-sided: $x[n] = 0$ for $n < N_1$ (causal if $N_1 \geq 0$)
• Left-sided: $x[n] = 0$ for $n > N_2$
• Two-sided: Extends infinitely in both directions

Periodic vs Aperiodic

A discrete signal is periodic with period $N$ if $x[n + N] = x[n]$ for all $n$, and $N$ is the smallest such positive integer.

Important: A discrete sinusoid $\cos(\omega_0 n)$ is periodic only if $\omega_0/(2\pi)$ is a rational number $M/N$, in which case the period is $N$ (with $M/N$ in lowest terms).

Symmetric (Even) and Antisymmetric (Odd)

• Even: $x[-n] = x[n]$ (symmetric about $n = 0$)
• Odd: $x[-n] = -x[n]$ (antisymmetric, with $x[0] = 0$)

Any signal has a unique decomposition: $x[n] = x_e[n] + x_o[n]$ where:

$$x_e[n] = \frac{x[n] + x[-n]}{2}, \quad x_o[n] = \frac{x[n] - x[-n]}{2}$$

Causal and Anti-Causal

• Causal: $x[n] = 0$ for $n < 0$ (signal starts at or after $n = 0$)
• Anti-causal: $x[n] = 0$ for $n > 0$

Energy and Power of Discrete Signals

Energy

$$E = \sum_{n=-\infty}^{\infty} |x[n]|^2$$

If $E$ is finite, $x[n]$ is an energy signal.

Average Power

$$P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2$$

If $P$ is finite and nonzero, $x[n]$ is a power signal.

Examples

• $x[n] = (0.5)^n u[n]$: $E = \sum_{n=0}^{\infty} 0.25^n = 1/(1-0.25) = 4/3$ → energy signal
• $x[n] = \cos(\pi n/4)$: infinite energy, $P = 1/2$ → power signal
• $x[n] = n \cdot u[n]$ (ramp): infinite energy AND infinite power → neither

Basic Operations on Discrete Signals

Time Shifting

$y[n] = x[n - n_0]$: delays by $n_0$ samples (if $n_0 > 0$) or advances (if $n_0 < 0$).

Time Reversal

$y[n] = x[-n]$: reflects the signal about $n = 0$.

Time Scaling (Decimation and Interpolation)

• Downsampling by $M$: $y[n] = x[Mn]$ — keeps every $M$-th sample, discards others
• Upsampling by $L$: Insert $L-1$ zeros between each sample

Unlike continuous-time scaling, discrete-time compression (downsampling) causes aliasing if the signal isn't bandlimited to $\pi/M$.

Addition and Multiplication

• $y[n] = x_1[n] + x_2[n]$: sample-by-sample addition
• $y[n] = x_1[n] \cdot x_2[n]$: sample-by-sample multiplication (modulation)

Accumulation (Running Sum)

$$y[n] = \sum_{k=-\infty}^{n} x[k]$$

This is the discrete-time analog of integration. The system $y[n] = y[n-1] + x[n]$ is the accumulator.

Discrete vs Continuous: Key Differences

Worked Examples

Example 1: Find the energy of $x[n] = 3\delta[n] + 2\delta[n-1] - \delta[n-3]$.

Solution: $E = |3|^2 + |2|^2 + |-1|^2 = 9 + 4 + 1 = 14$

Example 2: Is $x[n] = \cos(n)$ periodic?

$\omega_0 = 1$ rad/sample. Check: $1/(2\pi) = 0.1592...$ which is irrational. Therefore, $x[n]$ is NOT periodic.

Example 3: Decompose $x[n] = \{1, \underset{\uparrow}{2}, 3\}$ into even and odd parts.

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

$x_e[n] = \{2, \underset{\uparrow}{2}, 2\}$ (average)

$x_o[n] = \{-1, \underset{\uparrow}{0}, 1\}$ (half-difference)

Key Takeaways

• Discrete signals $x[n]$ exist only at integer indices — the foundation of all digital processing
• Any signal equals a weighted sum of shifted impulses: $x[n] = \sum_k x[k]\delta[n-k]$
• Energy signals have finite $\sum|x[n]|^2$; power signals have finite nonzero average power
• Discrete sinusoids are periodic only when $\omega_0/(2\pi)$ is rational
• Frequency is unique only in $(-\pi, \pi]$ — the fundamental difference from continuous-time
• Operations include shifting, reversal, scaling (decimation/interpolation), and accumulation