Discrete Time Signals

In digital signal processing, all real-world analog signals must be represented in a form that digital systems can process. A discrete-time signal is a sequence of values defined only at specific, equally spaced instants of time. Unlike a continuous-time signal which exists at every moment, a discrete-time signal has values only at integer sample indices. This concept is fundamental to all DSP theory and is one of the most heavily weighted topics in GATE Electronics and Communication Engineering.

Core Concept: What is a Discrete-Time Signal

A discrete-time signal is written as x[n] (square brackets denoting discrete-time), where n is an integer index representing the sample number. The signal is not defined for non-integer values of n. This is fundamentally different from a continuous-time signal x(t) which exists for all real values of t. A discrete-time signal is obtained by sampling a continuous-time signal at regular intervals, or it may originate natively as a sequence (such as daily temperature readings or stock prices).

The relationship between the discrete-time index n and actual time t is: t = n x T_s, where T_s is the sampling period in seconds. The reciprocal of T_s is the sampling frequency F_s = 1/T_s in Hz. According to the Nyquist-Shannon sampling theorem, F_s must be at least twice the highest frequency component of the original signal to allow perfect reconstruction.

Standard Elementary Sequences

Unit Impulse Sequence

The unit impulse sequence delta[n] equals 1 when n = 0 and equals 0 for all other values of n. It is the discrete-time counterpart of the Dirac delta function but is much simpler: it is just the number 1 at n=0. The unit impulse is the identity element for discrete convolution: any sequence convolved with delta[n] returns the sequence itself.

Unit Step Sequence

The unit step sequence u[n] equals 1 for all n >= 0 and equals 0 for n < 0. The step sequence is related to the impulse sequence by: delta[n] = u[n] - u[n-1]. Conversely, u[n] is the running sum (accumulation) of delta[n]: u[n] = sum from k = -infinity to n of delta[k].

Real Exponential Sequence

The real exponential sequence is defined as x[n] = a^n for all n, where a is a real number. If |a| < 1, the sequence decays towards zero (stable). If |a| > 1, the sequence grows without bound (unstable). If a is negative, the sequence alternates in sign. Exponential sequences arise naturally as impulse responses of linear time-invariant (LTI) systems with real poles.

Signal Operations

Several operations can be performed on discrete-time sequences. Time shifting by k samples transforms x[n] to x[n-k]. A positive k shifts the sequence to the right (delay), and a negative k shifts it to the left (advance). Time reversal reflects the sequence about the origin, transforming x[n] to x[-n].

Amplitude scaling multiplies every sample by a constant: y[n] = A x x[n]. Addition of two sequences is sample-by-sample: y[n] = x1[n] + x2[n]. Multiplication is also sample-by-sample: y[n] = x1[n] x x2[n]. These operations form the building blocks for implementing digital filters and signal transformations.

Energy and Power of Discrete-Time Signals

The total energy of a discrete-time signal is defined as E = sum from n = -infinity to +infinity of |x[n]|^2. The average power is defined as P = limit as N approaches infinity of (1 / (2N+1)) x sum from n = -N to N of |x[n]|^2.

A signal is classified as an energy signal if its total energy E is finite and nonzero (0 < E < infinity), which implies that its average power P = 0. A signal is classified as a power signal if its average power P is finite and nonzero (0 < P < infinity), which implies that its total energy E = infinity. Signals cannot simultaneously be both energy and power signals. A purely decaying exponential (|a| < 1) is an energy signal; a periodic sequence or a unit step is a power signal.

Periodicity of Discrete-Time Signals

A discrete-time signal x[n] is periodic with period N if x[n+N] = x[n] for all n, where N is the smallest positive integer satisfying this condition. For a discrete-time sinusoid x[n] = A cos(omega_0 x n + phi), the signal is periodic only if omega_0 / (2 x pi) is a rational number p/q. In that case, the period N = q (the denominator in lowest terms). This is a critical difference from continuous-time sinusoids, which are always periodic.

Numerical Example

Computing energy and power helps classify a signal and understand its behavior over time. For a finite-duration or decaying sequence, energy is finite. For a periodic or step sequence, energy is infinite but power is finite. The classification matters for choosing appropriate analysis tools such as DTFT versus DFS.

Given:
  x[n] = (0.5)^n * u[n]
  This is a causal decaying exponential sequence with a = 0.5

Why this formula applies:
  Energy of a signal is the sum of squared magnitudes over all n.
  Since the signal is zero for n < 0 due to u[n], the sum starts at n = 0.

Formula:
  E = sum from n=0 to infinity of |x[n]|^2
    = sum from n=0 to infinity of (0.5)^(2n)
    = sum from n=0 to infinity of (0.25)^n
    = 1 / (1 - 0.25)  [geometric series, |r| < 1]

Substitution:
  E = 1 / (1 - 0.25)

Calculation:
  E = 1 / 0.75
  E ≈ 1.333 (finite)

  Average Power P = E / infinity = 0 (since E is finite)

Final Answer:
  Energy E = 4/3 ≈ 1.333 (finite)
  Average Power P = 0
  Classification: x[n] = (0.5)^n u[n] is an ENERGY SIGNAL.

Exam Tip: GATE frequently tests energy and power signal classification. The key rule: if the sum of |x[n]|^2 over all n converges to a finite value, it is an energy signal with P = 0. If the sum diverges but the time-averaged power converges, it is a power signal with E = infinity. For geometric series in energy calculation, use E = 1/(1 - r^2) for x[n] = r^n u[n] when |r| < 1, giving E = 1/(1 - a^2).

Quick Revision

• Discrete-time signal x[n] is defined only at integer n. Square brackets distinguish discrete from continuous x(t).
• Unit impulse delta[n] = 1 at n=0, 0 elsewhere. Unit step u[n] = 1 for n >= 0, 0 for n < 0.
• Energy: E = sum of |x[n]|^2. Power: P = limit of (1/(2N+1)) x sum of |x[n]|^2.
• Energy signal: E finite, P = 0. Power signal: P finite, E = infinity. A signal cannot be both.
• For x[n] = a^n u[n] with |a| < 1: Energy = 1/(1 - a^2). This is a decaying energy signal.
• Discrete-time sinusoid is periodic only if omega_0 / (2*pi) is rational. Period N equals the denominator of that rational number in lowest terms.
• GATE trap: A unit step sequence u[n] has infinite energy but finite power P = 1/2. Do not confuse it with an energy signal.