Introduction to Signals
Signals are functions that convey information about the behavior or attributes of phenomena. In electrical engineering, signals represent time-varying quantities such as voltage, current, or electromagnetic fields.
Figure 1: Various signal types including sine wave, square wave, and their frequency domain representations
Signal Analysis Calculator
Signal Characteristics:
- Amplitude: Magnitude of the signal
- Frequency: Rate of variation with time
- Phase: Position in the cycle relative to reference
- Bandwidth: Range of frequencies contained in the signal
- Duration: Time over which signal exists
Signal Classification
Continuous-Time Signals
- Definition: Defined for all time values
- Mathematical Form: x(t), t ∈ ℝ
- Examples: Sine wave, analog signals
- Applications: Analog circuits, sensors
Discrete-Time Signals
- Definition: Defined at specific time instants
- Mathematical Form: x[n], n ∈ ℤ
- Examples: Sampled signals, digital data
- Applications: Digital systems, DSP
Analog Signals
- Definition: Continuous amplitude and time
- Range: Infinite possible values
- Examples: Audio signals, temperature
- Advantages: Infinite resolution
Digital Signals
- Definition: Discrete amplitude and time
- Range: Finite set of values
- Examples: Binary data, computer files
- Advantages: Noise immunity, processing
Mathematical Representation of Signals
Sinusoidal Signals
x(t) = A × sin(ωt + φ)
Where:
- A = Amplitude (peak value)
- ω = Angular frequency = 2πf (rad/s)
- f = Frequency (Hz)
- φ = Phase angle (radians)
- t = Time (seconds)
Sinusoidal Signal Analysis
Given: x(t) = 5 × sin(100πt + π/3)
Identify parameters:
Amplitude A = 5
Angular frequency ω = 100π rad/s
Frequency f = ω/(2π) = 50 Hz
Phase φ = π/3 = 60°
Amplitude A = 5
Angular frequency ω = 100π rad/s
Frequency f = ω/(2π) = 50 Hz
Phase φ = π/3 = 60°
RMS Value:
V_rms = A/√2 = 5/√2 = 3.54
V_rms = A/√2 = 5/√2 = 3.54
Period:
T = 1/f = 1/50 = 0.02 seconds = 20 ms
T = 1/f = 1/50 = 0.02 seconds = 20 ms
Complex Exponential Signals
Complex exponentials are fundamental in signal analysis:
e^(jωt) = cos(ωt) + j × sin(ωt)
Euler's formula relates complex exponentials to trigonometric functions
Complex Exponential Calculation
Given: e^(jπ/4)
Solution using Euler's formula:
e^(jπ/4) = cos(π/4) + j × sin(π/4)
= √2/2 + j × √2/2
= 0.707 + j0.707
e^(jπ/4) = cos(π/4) + j × sin(π/4)
= √2/2 + j × √2/2
= 0.707 + j0.707
Magnitude and phase:
|e^(jπ/4)| = √(0.707² + 0.707²) = 1
∠e^(jπ/4) = π/4 = 45°
|e^(jπ/4)| = √(0.707² + 0.707²) = 1
∠e^(jπ/4) = π/4 = 45°
Unit Impulse and Unit Step Functions
Unit Impulse (Dirac Delta)
δ(t) = { ∞, t = 0
0, t ≠ 0 }
0, t ≠ 0 }
Properties:
- ∫δ(t)dt from -∞ to ∞ = 1
- x(t) * δ(t) = x(t) (sifting property)
Unit Step Function
u(t) = { 1, t ≥ 0
0, t < 0 }
0, t < 0 }
Relationship: δ(t) = du(t)/dt
Signal Energy and Power
Energy Signals
Energy E = ∫|x(t)|² dt from -∞ to ∞
Finite energy, zero average power
Power Signals
Average Power P = lim(T→∞) [1/(2T)] × ∫|x(t)|² dt from -T to T
Finite average power, infinite energy
Signal Energy Calculation
Given: x(t) = A × e^(-αt) × u(t), α > 0
Energy calculation:
E = ∫₀^∞ |A × e^(-αt)|² dt
E = A² × ∫₀^∞ e^(-2αt) dt
E = A² × [1/(2α)]
E = ∫₀^∞ |A × e^(-αt)|² dt
E = A² × ∫₀^∞ e^(-2αt) dt
E = A² × [1/(2α)]
Special case: A = 1, α = 0.5
E = 1² × 1/(2 × 0.5) = 1
E = 1² × 1/(2 × 0.5) = 1