Exponential Signals (Real and Complex)

Main ideas / concepts covered

• Goal of the lecture: Study real and complex exponential signals, and plot their waveforms using MATLAB.

• Starting signal definition (real/complex exponential):

  • [ x(t) = a e^{st} ]

  • For simplicity, take (a=1):

    • [ x(t) = e^{st} ]

  • Let (s) be complex:

    • [ s=\Sigma + j\Omega ]

    • where (\Sigma) is the real part and (\Omega) is the imaginary part (interpreted as angular frequency).

Rewrite using Euler’s formula

• [ x(t)=e^{(\Sigma + j\Omega)t}=e^{\Sigma t}\cdot e^{j\Omega t} ]

• Using Euler’s formula:

  • [ e^{jx}=\cos x + j\sin x ]

• Therefore:

  • [ x(t)=e^{\Sigma t}\left(\cos(\Omega t)+j\sin(\Omega t)\right) ]

Case analysis by (\Omega) and (\Sigma)

• Real exponential signals occur when (\Omega=0):

  • Since (\cos(0)=1) and (\sin(0)=0):

    • [ x(t)=e^{\Sigma t} \quad \text{(purely real)} ]

  • Split by (\Sigma):

    • (\Sigma>0): exponentially rising
    • (\Sigma<0): exponentially decaying

• Complex exponential signals occur when (\Omega\neq 0):

  • (\Sigma=0): purely sinusoidal form
    • Real part: (\cos(\Omega t))
    • Imag part: (\sin(\Omega t))
  • (\Sigma>0): exponentially rising (overall magnitude grows)
    • Real part: (e^{\Sigma t}\cos(\Omega t))
    • Imag part: (e^{\Sigma t}\sin(\Omega t))
  • (\Sigma<0): exponentially decaying (overall magnitude shrinks)
    • Real part: (e^{\Sigma t}\cos(\Omega t))
    • Imag part: (e^{\Sigma t}\sin(\Omega t))

Visualization emphasis

For complex exponentials, the lecturer emphasizes 3D plotting because textbooks may show only 2D projections.

• The real and imaginary parts can be shown as 2D plots.
• The full complex signal (x(t)) is shown as a 3D waveform; in 2D it may look like a sinusoid depending on the viewpoint.

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MATLAB plotting methodology (as given in the lecture)

A) Plot real exponential signal ((\Omega = 0))

The lecture plots two cases together: rising ((\Sigma>0)) and decaying ((\Sigma<0)).

1. Open MATLAB editor

   • Press Ctrl + N.

2. Define time vector (t)

   • Set (t) from (-2) to (2)
   • Step size: 0.01

3. Rising case: choose (\Sigma = 2)

   • Define:
     • [ x(t)=e^{\Sigma t} ]

4. Plot (x(t))

   • Use MATLAB plot with:
     • independent variable first (t), then function (X)
   • Styling:
     • Color red (“R” in the plot call)
     • Line width using LineWidth set to 2.5

5. Add labels

   • Title: “real exponential signal”
   • X-axis: time (t) (shown as “t”)
   • Y-axis: (x(t))

6. Run code

   • Press the Run button.

7. Decaying case on the same figure

   • Choose (\Sigma_1=-2) (negative)

   • Define:

     • [ x_1(t)=e^{\Sigma_1 t} ]

   • Add second curve to the existing plot:

     • Plot both curves using plot(t, X1, ...)
     • Color green (“G”)

8. Observed results

   • At (t=0), both exponentials equal 1
   • For (t>0):
     • (\Sigma>0) rises (grows exponentially)
     • (\Sigma<0) decays (shrinks exponentially)

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B) Plot complex exponential signals ((\Omega \neq 0))

• The lecturer emphasizes that complex exponentials are more important and typically require 3D plotting.
• Three cases described (no detailed MATLAB code provided in the subtitles):

1. (\Sigma=0)

   • [ x(t)=\cos(\Omega t)+j\sin(\Omega t) ]

   • Real part: 2D sinusoid

   • Imag part: 2D sinusoid
   • Full complex signal: shown as 3D

2. (\Sigma>0)

   • [ x(t)=e^{\Sigma t}\left(\cos(\Omega t)+j\sin(\Omega t)\right) ]

   • Real and imaginary parts: rise exponentially

   • Full signal: 3D exponentially rising

3. (\Sigma<0)

   • Same form as above, but (\Sigma) is negative
   • Real and imaginary parts: decay exponentially
   • Full signal: 3D exponentially decaying

Textbooks may show a 2D representation that can resemble a projection of the 3D behavior.

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Speakers / sources featured

• Unidentified lecturer / course instructor (speaking throughout the video)
• No other explicitly named sources (MATLAB is mentioned; Euler’s formula is referenced mathematically)