Continuous and Discrete-Time Sine Signals

Continuous-Time Sine Signals

• A simple harmonic oscillation is mathematically described by xₐ(t) = A cos(Ωt + θ), where -∞ < t < ∞.
• Ω represents 2πF, measured in rad/s.
• A is the amplitude.
• F is frequency, measured in Hz.
• θ is the phase, measured in radians.
• Analog sinusoidal signals are characterized by periodicity for every fixed value of the frequency F.
• xₐ(t + Tₚ) = xₐ(t), where Tₚ = 1/F is the fundamental period of the sinusoidal signal.
• Continuous-time sinusoidal signals with distinct frequencies are themselves distinct.
• Increasing the frequency F increases the rate of oscillation, including more periods within a given time interval.

Discrete-Time Sine Signals

• A discrete-time sinusoidal signal may be expressed as x(n) = A cos(ωn + θ), where -∞ < n < ∞.
• n is an integer variable representing the sample number.
• A is the amplitude of the sinusoid.
• ω is the frequency in radians per sample.
• θ is the phase in radians
• Expressing the equation with ω = 2πf gives x(n) = A cos(2πfn + θ).
• Discrete-time sinusoids differ from continuous-time in the following ways:
• A discrete-time sinusoid is periodic only if its frequency f is a rational number.
• Discrete-time sinusoids with frequencies separated by an integer multiple of 2π are identical.
• The highest oscillation rate in a discrete-time sinusoid is attained when ω = π (or ω = -π), or equivalently, f = 1/2 (or f = -1/2).
• A discrete-time signal x(n) is periodic with period N if and only if x(n + N) = x(n) for all n.
• The smallest value of N for which the equation above holds true is the fundamental period.
• For a sinusoid with frequency f₀ to be periodic, cos[2πf₀(n + N) + θ] = cos(2πf₀n + θ).
• This relation is true if and only if there exists an integer k such that 2πf₀N = 2πk.
• Equivalently, f₀ = k/N.
• A discrete-time sinusoidal signal is periodic only if its frequency f₀ can be expressed as the ratio of two integers (i.e., f₀ is rational).
• If f₁ = 31/60, then k₁ = 31 and N₁ = 60.
• If f₂ = 30/60, simplify to f₂ = 1/2 before deriving k₂ (= 1) and N₂ (= 2).
• Even if f₁ and f₂ are almost identical, their periods N₁ and N₂ can be completely different.
• Discrete-time sinusoids with frequencies separated by an integer multiple of 2π are identical; cos[(ω₀ + 2π)n + θ] = cos[ω₀n + 2πn + θ] = cos(ω₀n + θ).
• All sinusoidal sequences xₖ(n) = A cos(ωₖn + θ), where k = ±0,1,2,..., are identical and indistinguishable, with ωₖ = ω₀ + 2kπ, and -π ≤ ω₀ ≤ π.
• Any sequence from a sinusoid with frequency |ω| > π, or |f| > 1/2, is identical to a sequence from a sinusoidal signal with frequency |ω| < π.
• The sinusoid having the frequency |ω| > π is an alias of a corresponding sinusoid with frequency |ω| < π.
• Frequencies in the range -π ≤ ω ≤ π, or -1/2 ≤ f ≤ 1/2, are unique, and all frequencies |ω| > π, or |f| > 1/2, are aliases.
• The highest oscillation rate in a discrete-time sinusoid is attained when ω = π (or ω = -π), or f = 1/2 (or f = -1/2).
• When ω₀ varies from 0 to π, values of ω₀ = 0, π/8, π/4, π/2, π correspond to f = 0, 1/16, 1/8, 1/4, 1/2, resulting in periodic sequences with periods N = ∞, 16, 8, 4, 2.
• Since discrete-time sinusoidal signals with frequencies separated by an integer multiple of 2π are identical, frequencies in any interval ω₁ ≤ ω ≤ ω₁ + 2π constitute all existing discrete-time sinusoids or complex exponentials.
• The frequency range for discrete-time sinusoids is finite with duration 2π.
• Usually, the range 0 ≤ ω ≤ 2π or -π ≤ ω ≤ π or (-1/2 ≤ f ≤ 1/2) is chosen, as the fundamental range.