Communication & Engineering

Switch vs. Router

Periodic Signals VS Nonperiodic (Aperiodic) Signals

What is Peak Amplitude?

The peak amplitude of a signal is the absolute value of its highest intensity, representing the maximum energy it carries. In simple terms, it is the highest point (crest) or the lowest point (trough) a signal reaches from its baseline or zero level.

Key Characteristics

• Symbol of Energy: It indicates the amount of energy or power a signal possesses. The higher the amplitude, the more energy the signal carries.
• Unit of Measurement: For electric signals, peak amplitude is typically measured in Volts (V).
• Physical Effect:
  • In audio, higher amplitude means a louder sound.
  • In light, higher amplitude means a brighter light.

Same Phase and Frequency

When two signals have the same phase and frequency but different amplitudes:

• They start at the exact same time.
• They reach their maximum and minimum points (peaks and valleys) at the same time.
• The only difference is the height of the waves—one signal is “stronger” or “taller” than the other.

Different types of Units

The two extreme cases of signal frequency

1. Zero Frequency

• Condition: If a signal does not change at all over time and maintains a constant voltage level.
• Logic: Frequency measures how many times a signal completes a pattern or cycle within a specific time. Since the signal is static and unchanging, it never completes a cycle.
• Result: In this case, the frequency is 0 Hz.
• Example: A battery’s voltage (Direct Current or DC), which remains constant over time.

2. Infinite Frequency

• Condition: If a signal changes instantaneously (e.g., jumping from 0V to 5V in zero time).
• Logic: Frequency is the inverse of the period ($f = 1/T$). If the change happens in “no time,” the period ($T$) is 0.
• Calculation: Mathematically, $1/0$ is considered infinite or unbounded.
• Result: In this case, the frequency is Infinite.
• Example: An ideal digital pulse that rises perfectly vertically.

The Bottom Line: Frequency is essentially a measure of how fast a signal changes. No change means zero frequency, while an instant change means infinite frequency.

Sine and cos and tan graph

The Sine Wave ($\sin$)

The Sine wave is the “standard” starting wave.

• Starting Point (at $0^\circ$): Starts at zero.
• Direction: Goes Up (Positive) immediately.
• Key Property: It is at its peak at $90^\circ$ and back to zero at $180^\circ$.
• Equation: $y = A \sin(2\pi ft + \phi)$

The Cosine Wave ($\cos$)

The Cosine wave is actually just a Sine wave that has a head start!

• Starting Point (at $0^\circ$): Starts at the Peak (Maximum).
• Direction: Goes Down immediately.
• The 90° Rule: A Cosine wave is identical to a Sine wave with a $90^\circ$ phase shift.
  • $\cos(x) = \sin(x + 90^\circ)$
• Key Property: It hits zero at $90^\circ$ and hits its lowest point (negative peak) at $180^\circ$.

The Tangent Wave ($\tan$)

The Tangent wave is completely different from Sine and Cosine because it is not a “smooth” repeating loop. It is defined as $\tan = \frac{\sin}{\cos}$.

• Starting Point (at $0^\circ$): Starts at zero.
• Shape: It looks like a vertical curve that shoots up to infinity.
• The “Infinite” Criteria:
  • Since $\tan = \frac{\sin}{\cos}$, whenever Cosine is zero, the Tangent wave becomes infinite (Vertical Asymptote).
  • This happens at $90^\circ$ and $270^\circ$.
• Key Property: It does not have a “Peak Amplitude” because it goes to infinity.
  

Three sine waves with the same amplitude and frequency, but different phases

The time-domain and frequency-domain plots of a sine wave

Amplitude for periodic and nonperiodic

The time and frequency domains of periodic and nonperiodic digital signals

This image explains the relationship between the Time Domain and Frequency Domain for digital signals. It highlights two key scenarios:

1. Figure (a): Periodic Digital Signal

• Time Domain (Left): The signal repeats itself at regular intervals (e.g., a perfect square wave).
• Frequency Domain (Right): When viewed in the frequency domain, the signal is represented by discrete vertical lines.
  • This means the signal is composed of a specific set of frequencies (e.g., $f, 3f, 5f$, etc.).
  • This mathematical representation is known as a Fourier Series.
  • Note that for a standard square wave, these frequencies are usually odd harmonics of the fundamental frequency.

2. Figure (b): Non-periodic Digital Signal

• Time Domain (Left): The signal does not repeat; it is a single pulse (which often represents a single bit of data in telecommunications).
• Frequency Domain (Right): Instead of discrete lines, we see a continuous curve.
  • This indicates that a non-periodic signal contains an infinite range of frequencies packed closely together.
  • This mathematical representation is known as a Fourier Transform.

Transmit a digital signal by using

1. baseband transmission
2. broadband transmission(using modulation)

Baseband Transmission

Baseband transmission means sending a digital signal over a channel without changing the digital signal to an analog signal that is baseband transmission.

a low-pass channel,

a channel with a bandwidth that starts from zero.

Low-Pass Channel with Limited Bandwidth