Elementary Discrete-Time Sequences

The Building Blocks

Every discrete-time signal can be built from a small set of elementary sequences. These are the discrete cousins of the impulse, step, sine, and exponential functions from continuous-time calculus.

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1. Unit Sample (Unit Impulse)

\[\delta[n] = \begin{cases} 1, & n=0 \\ 0, & n \ne 0 \end{cases}\]

δ[n]
 1 |  *
   |
 0 +--+--+--+--+--+-- n
   -2 -1  0  1  2

The shifted impulse $\delta[n-k]$ is 1 only at $n=k$.

The Sifting Property

Any sequence can be written as a sum of scaled impulses:

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

This identity is the foundation of convolution: feed a system impulses, scale, and add.

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2. Unit Step

\[u[n] = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \end{cases}\]

u[n]
 1 |  *  *  *  *  *
   |
 0 +--+--+--+--+--+-- n
   -2 -1  0  1  2

Relationships

\[\delta[n] = u[n] - u[n-1] \qquad u[n] = \sum_{k=-\infty}^{n} \delta[k]\]

A shifted step $u[n-k]$ turns on at $n=k$.

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3. Unit Ramp

\[r[n] = n\,u[n] = \begin{cases} n, & n \ge 0 \\ 0, & n < 0 \end{cases}\]

r[n]
   |        *
 2 |     *
 1 |  *
 0 +--+--+--+--+-- n
   0  1  2  3  4

The ramp is the running sum of the step: $r[n] = \sum_{k=-\infty}^{n} u[k]$.

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4. Real Sinusoidal Sequence

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

• $A$ = amplitude
• $\omega_0$ = angular frequency in radians/sample
• $\phi$ = phase in radians

When obtained by sampling $A\cos(2\pi F t + \phi)$ at $F_s$, the digital frequency is

\[\omega_0 = 2\pi \frac{F}{F_s}\]

so $\omega_0$ is dimensionless (rad/sample), unlike the analog $\Omega = 2\pi F$ (rad/s).

Periodicity: The Discrete Difference

A discrete sinusoid is periodic only if there exists an integer $N>0$ such that

\[\omega_0 N = 2\pi m \quad \text{for some integer } m\]

Equivalently, $x[n]=A\cos(\omega_0 n)$ is periodic iff

\[\frac{\omega_0}{2\pi} = \frac{m}{N} \text{ is rational}\]

Then the fundamental period is $N = \frac{2\pi}{\gcd(\omega_0,2\pi)}$ in integer form.

Sequence	$\omega_0/2\pi$	Periodic?	Period
$\cos(\pi n/4)$	$1/8$	Yes	8
$\cos(0.1\pi n)$	$1/20$	Yes	20
$\cos(0.3n)$	irrational	No	—
$\cos(n)$	irrational	No	—

Exam trap: every continuous sinusoid is periodic, but not every discrete sinusoid is. Irrational digital frequency gives a non-periodic sequence.

Aliasing of Discrete Frequencies

Discrete frequencies are unique only over an interval of $2\pi$. In fact

\[\cos[(\omega_0 + 2\pi k)n] = \cos(\omega_0 n)\]

so $\omega_0$ and $\omega_0 + 2\pi$ are the same discrete sinusoid. Frequencies also fold: $\omega$ and $2\pi - \omega$ have the same cosine samples.

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5. Complex Exponential Sequence

\[x[n] = A e^{j(\omega_0 n + \phi)} = A\cos(\omega_0 n+\phi) + jA\sin(\omega_0 n+\phi)\]

Euler’s relation is the bridge between the two forms. Complex exponentials are the eigenfunctions of LTI systems:

\[e^{j\omega n} \;\longrightarrow\; H(e^{j\omega})\,e^{j\omega n}\]

That single fact is why the DTFT exists.

Harmonically Related Complex Exponentials

For a period-$N$ sequence, the set

\[e^{j2\pi k n/N}, \quad k = 0, 1, \dots, N-1\]

contains exactly $N$ distinct sequences and forms an orthogonal basis for all period-$N$ sequences. This is the discrete Fourier series, and it is the reason the DFT has length $N$.

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6. Real Exponential Sequence

\[x[n] = A\alpha^n, \quad -\infty < n < \infty\]

$\alpha$	Behaviour
$\alpha > 1$	Growing exponential
$0 < \alpha < 1$	Decaying exponential
$\alpha = 1$	Constant
$\alpha = -1$	Alternating $\pm A$
$-1 < \alpha < 0$	Decaying alternating
$\alpha < -1$	Growing alternating

A common case is $\alpha = e^{\sigma_0}$:

\[x[n]=A e^{(\sigma_0+j\omega_0)n} = A e^{\sigma_0 n}\cos(\omega_0 n) + jA e^{\sigma_0 n}\sin(\omega_0 n)\]

The real part is a damped sinusoid — the impulse response of a stable resonator.

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Summary Table

Sequence	Definition	Z-transform	ROC
$\delta[n]$	$1$ at $n=0$	$1$	all $z$
$u[n]$	$1$ for $n\ge 0$	$\frac{1}{1-z^{-1}}$	$\vert z\vert>1$
$a^n u[n]$	exponential	$\frac{1}{1-az^{-1}}$	$\vert z\vert>\vert a\vert$
$n a^n u[n]$	ramp-like	$\frac{az^{-1}}{(1-az^{-1})^2}$	$\vert z\vert>\vert a\vert$
$\cos(\omega_0 n)u[n]$	sinusoid	$\frac{1-\cos\omega_0 z^{-1}}{1-2\cos\omega_0 z^{-1}+z^{-2}}$	$\vert z\vert>1$

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Operations on Sequences

The seven basic operations split into two groups:

Amplitude (y-axis): addition, subtraction, multiplication (windowing/modulation), amplitude scaling.

Time (x-axis): shifting (delay/advance), reversal (folding), scaling (down/up-sampling).

Sequences of different lengths can be added or multiplied after zero-padding the shorter one to the same index range.

Example: $a[n]={3,4,6,-9,0}$ and $b[n]={2,-1,4,5,-3}$ for $0\le n\le 4$.

Operation	Result
$a[n]+b[n]$	${5,3,10,-4,-3}$
$a[n]b[n]$	${6,-4,24,-45,0}$
$1.5a[n]$	${4.5,6,9,-13.5,0}$