Lecture Notes: Signals and Systems

Introduction

• The lecture discusses basic signals, both continuous and discrete time.
• Focus on continuous time sinusoidal signals and their properties.

Continuous Time Sinusoidal Signal

• Mathematical Expression:
  [ x(t) = A \cos(\omega_0 t + \phi) ]
  • Parameters:
    • A: Amplitude
    • ( \omega_0 ): Frequency
    • ( \phi ): Phase
• Graphical Representation:
  • Displayed sinusoidal waveform.

Properties of Sinusoidal Signals

• Periodicity:

  • A sinusoidal signal is periodic, meaning it repeats over time.
  • Defined period:
    • ( T = \frac{2\pi}{\omega_0} )
  • Demonstrated mathematically by substituting time shifts into the sinusoidal equation.

• Time Shift vs Phase Change:

  • Time shift of a sinusoid leads to a corresponding phase shift:
    • ( t + t_0 ) implies phase change of ( \omega_0 t_0 )
  • Example: Sinusoidal signal shifted by ( t_0 = \frac{T}{4} ) results in a phase change of ( -\frac{\pi}{2} ).

Symmetry of Sinusoidal Signals

• Even and Odd Symmetry:
  • Even Signal:
    • Symmetric about the origin:
      • ( x(t) = x(-t) )
  • Odd Signal:
    • Anti-symmetric:
      • ( x(t) = -x(-t) )

Discrete Time Sinusoids

• Mathematical Expression:
  [ x[n] = A \cos(\omega_0 n + \phi) ]
  • Similar properties to continuous time but with integer values for n.

Time Shift and Phase Change in Discrete Time

• Time shifts in discrete signals lead to phase changes, but reverse is not always true.
• Must consider periodicity based on whether ( \frac{2\pi}{\omega_0} ) can yield an integer.

Periodicity in Discrete Time

• Not all discrete time sinusoids are periodic.
• For periodicity, ( \omega_0 N = 2\pi m ) must hold true.
  • Example:
    • ( \omega_0 = \frac{2\pi}{12} ) is periodic with a period of 12.
    • ( \omega_0 = \frac{1}{6} ) is not periodic since ( 12\pi ) is not an integer.

Differences Between Continuous and Discrete Time Sinusoids

1. Time Shift and Phase Change:
   • Continuous: time shift and phase change equivalent.
   • Discrete: one-way relationship.
2. Periodic Nature:
   • Continuous: all sinusoids are periodic.
   • Discrete: periodicity depends on frequency.
3. Frequency Representation:
   • Discrete signals repeat every 2( \pi ) interval. Continuous does not.

Real and Complex Exponentials

• Real Continuous Time Exponential:
  [ x(t) = c e^{at} ]

  • Positive a: growing; negative a: decaying.

• Real Discrete Time Exponential:
  [ x[n] = c \alpha^n ]

  • Varies based on alpha's value (greater than or less than 1).

• Complex Continuous Time Exponential:
  [ x(t) = c e^{at} ]

  • c and a can be complex.
  • Utilizes Euler’s relation to relate to sinusoids.

Summary of Key Points

• Importance of sinusoidal signals as building blocks for signal analysis.
• Differences between continuous and discrete representations.
• Next lecture will cover step signals and impulse signals, essential for signal and system analysis.