Signals and Systems Lecture - Real and Complex Exponential Signals

Importance of Exponential Signals

• Exponential signals are crucial for the study of signals and systems as they solve differential equations describing many systems.
• Exponential signals will recur throughout the course.

Continuous Time Real Exponential Signal

• Mathematical Representation:
  • x(t) = c * e^(a * t)
  • c = positive constant
  • a = real number, can be positive or negative
• Cases:
  • a > 0: Growing exponential function
  • a < 0: Decaying exponential function

Example Activity

• Sketch x(t) = 5 * e^(-t)
  • Passes through 5 at t=0
  • Decays to 0 as t increases*

Complex Exponential Signals

• Requires a review of complex numbers:
  • Complex Number Forms:
    • Polar Form: z = r * e^(j*θ)
      • r = magnitude
      • θ = phase angle
    • Rectangular Form: z = a + j*b
      • a = real part
      • b = imaginary part
  • Complex Plane: Representation of z as a point or vector*

Euler’s Formula

• Formula: e^(jθ) = cos(θ) + jsin(θ)

Special Cases

• Unit Circle (r=1): Special values for simplifying complex exponentials
  • e^(j*2π) = 1
  • e^(j*π/2) = j
  • e^(-j*π/2) = -j
  • e^(jπ) = e^(-jπ) = -1*

Continuous Time Complex Exponential Signals

• General Form: x(t) = C * e^(σ + jω)t
  • C = complex constant in polar form
  • σ = real part of exponent (related to time constant)
  • ω = imaginary part of exponent (radian frequency)
• Oscillation:
  • Real Part: Oscillates as cosine function
  • Imaginary Part: Oscillates as sine function*

Example Activity

• Given x(t) = 5 * cos(6πt)
  • Amplitude = 5
  • Radian Frequency (ω) = 6π radians/sec
  • Physical Frequency (f) = 3 Hz
  • Period (T) = 1/3 sec*

Discrete Time Exponential Signals

Real Exponential Sequences

• Mathematical Representation: x(n) = c * α^n
  • α > 1: Exponential growth
  • 0 < α < 1: Exponential decay
  • α < 0: Alternating sequence growth or decay*

Example Activity

• Sketch x(n) = (1/2)^n for n = -2 to 2

Imaginary Exponential Sequences

• Form: x(n) = e^(jωn)
  • Using Euler’s formula: x(n) = cos(ωn) + jsin(ω*n)
  • Oscillates: Real part as cosine, imaginary part as sine*

Frequency and Periodicity in Discrete Time

• Frequency (ω): Angle in radians, bounded by [0, 2π)
• Periodicity Requirement: Sequence is periodic if it satisfies specific integer multiples
  • Equation: N = 2π*m/ω where m is the smallest integer
  • Example: x(n) = cos(7π*n/3), ω = 7π/3, Period = 6](streamdown:incomplete-link)

Discrete Time Frequency as Radians per Sample

• Concept: x(n) = sample version of x(t)
• Radians per Sample: ω_d = ω * T_s
  • Example: 2 MHz signal, sampled at 10 million samples/sec, results in discrete frequency of 1.26 radians/sample*

This concludes part two of the lecture. We will continue in part three.