Discrete Time Signals

Discrete-time signals are sequences defined only at integer-valued time indices n, and are undefined between those integers. They arise naturally in digital signal processing, where a microprocessor stores and manipulates data as arrays of numbers representing audio samples, image pixels, or sensor readings.

Core Concept

A discrete-time signal x[n] is defined only for integer values of the index n. The notation uses square brackets to distinguish from the round-bracket notation x(t) for continuous-time signals. Between integers, x[n] is simply not defined; there is no interpolation implied.

The two most fundamental discrete-time signals are the unit impulse and the unit step. The unit impulse delta[n] equals 1 at n = 0 and 0 everywhere else. The unit step u[n] equals 1 for n >= 0 and 0 for n < 0. Every finite-duration sequence can be written as a weighted sum of shifted impulses: x[n] = sum_k x[k]*delta[n-k].

Energy and power classification follow the same logic as for continuous-time signals, but the integral is replaced by a sum. A discrete-time sinusoid A*cos(omega_0*n + phi) is periodic only if omega_0/(2*pi) is a rational number. This is different from continuous-time sinusoids, which are always periodic.

Key Formula

Energy and power of x[n]:

Energy: E = sum_{n=-inf}^{inf} |x[n]|^2

Power: P = lim_{N->inf} (1/(2N+1)) * sum_{n=-N}^{N} |x[n]|^2

Periodicity condition for x[n] = A*cos(omega_0*n): x[n] is periodic if and only if omega_0 = 2*pi*p/q for integers p and q. The fundamental period is then N = q/gcd(p,q).

Problem: Determine if x[n] = cos(0.3*pi*n) is periodic. If yes, find the period N.

Step 1: Identify omega_0
  omega_0 = 0.3*pi

Step 2: Check if omega_0 / (2*pi) is rational
  omega_0 / (2*pi) = 0.3*pi / (2*pi) = 0.3/2 = 0.15 = 3/20
  This is rational (p=3, q=20).

Step 3: Find the fundamental period
  N = q / gcd(p, q) = 20 / gcd(3, 20) = 20 / 1 = 20

Step 4: Verify
  x[n+20] = cos(0.3*pi*(n+20)) = cos(0.3*pi*n + 6*pi)
           = cos(0.3*pi*n)  [since 6*pi is a multiple of 2*pi]
  Confirmed periodic.

Final Answer: x[n] = cos(0.3*pi*n) is periodic with fundamental period N = 20.

Exam Tip: A discrete-time sinusoid cos(omega_0*n) is periodic only if omega_0/(2*pi) is rational. If it is irrational (for example omega_0 = 1 rad/sample), the sequence is NOT periodic. This differs from continuous-time where every sinusoid is periodic. GATE often tests this distinction. Also remember: x[n] = a^n * u[n] is an energy signal only if |a| < 1.

Properties Summary

• Defined only for integer n: x[n] is a sequence, not a function of real t.
• Unit impulse: delta[n] = 1 at n=0, 0 elsewhere. Sifting: sum_n x[n]*delta[n-k] = x[k].
• Unit step: u[n] = 1 for n>=0, 0 for n<0. Relation: delta[n] = u[n] - u[n-1].
• Energy: E = sum |x[n]|^2; signal is an energy signal if E is finite.
• Periodicity: cos(omega_0*n) is periodic iff omega_0/(2*pi) is rational.
• Even/Odd: x_e[n] = [x[n]+x[-n]]/2, x_o[n] = [x[n]-x[-n]]/2. x[n] = x_e[n]+x_o[n].

Quick Revision

• Square brackets x[n] mean discrete-time; round brackets x(t) mean continuous-time.
• Every sequence: x[n] = sum_k x[k]*delta[n-k], built from shifted impulses.
• Geometric sequence a^n*u[n]: energy signal iff |a| < 1.
• Periodicity needs rational omega_0/2pi. Fundamental period N = smallest positive integer satisfying x[n+N] = x[n].
• u[n] = sum_{k=0}^{inf} delta[n-k]. Accumulation of impulses gives the step.
• DT sinusoids with omega_0 = pi and omega_0 = -pi are identical. DT frequency is unique only in a 2*pi range.
• Exam trap: assuming a discrete-time sinusoid is always periodic. It is periodic only when omega_0/(2*pi) is rational.