If you could distill the essence of any analog signal down to its most fundamental properties, you would arrive at two quantities: amplitude and frequency. These are the twin pillars upon which all analog communication rests.

Amplitude tells us how strong a signal is—whether it's a whisper or a shout, a gentle ripple or a towering wave. Frequency tells us how fast the signal oscillates—whether it's the slow rumble of bass or the rapid vibration of a soprano's highest note.

Together, amplitude and frequency describe the essential character of any periodic signal. More importantly, they are the properties that engineers manipulate to encode information for transmission. Amplitude Modulation (AM), Frequency Modulation (FM), and their digital counterparts all work by systematically varying these fundamental properties to carry data.

In this page, we will explore amplitude and frequency with the depth they deserve—from fundamental definitions through mathematical relationships to practical engineering applications.

Amplitude is the measure of a signal's magnitude—how large the signal is at any given instant or characteristic point. For a sinusoidal wave, amplitude represents the maximum displacement from the equilibrium (zero) position. However, there are multiple ways to characterize amplitude, each serving different purposes.

Instantaneous Amplitude:

The value of a signal at any specific instant in time. For a sinusoid:

v(t) = A sin(2πft + φ)

The instantaneous amplitude is v(t), which varies continuously from -A to +A.

Peak Amplitude (Vp):

The maximum absolute value the signal attains. For a symmetric sinusoid with no DC offset:

Vp = A

This is the distance from zero to the maximum (or minimum) of the waveform.

Peak-to-Peak Amplitude (Vpp):

The total excursion from minimum to maximum:

Vpp = Vmax - Vmin = 2A (for symmetric signals)

Peak-to-peak is what you measure directly on an oscilloscope. It's the most visually intuitive amplitude measurement.

Average Amplitude:

The mean value over one complete cycle. For a pure sinusoid with no DC offset:

Vavg = 0 (positive and negative halves cancel)

For a full-wave rectified sinusoid:

Vavg = 2Vp/π ≈ 0.637 × Vp

The Root Mean Square (RMS) value deserves special attention because it connects amplitude to power—the fundamental quantity in signal transmission.

Mathematical Definition:

The RMS value of a periodic function v(t) with period T is:

Vrms = √[(1/T) ∫₀ᵀ v²(t) dt]

In words: square the signal, average over one period, take the square root.

For a Sinusoid:

v(t) = Vp sin(ωt)

v²(t) = Vp² sin²(ωt) = Vp² × (1 - cos(2ωt))/2

Average of sin²(ωt) over one period = 1/2

Therefore: Vrms = Vp/√2 ≈ 0.707 × Vp

Power in Resistive Loads:

For a resistor R:

• Instantaneous power: P(t) = v²(t)/R
• Average power: Pavg = Vrms²/R = Vp²/(2R)

This is why RMS is definitive for power calculations—it directly gives the DC-equivalent power level.

Why the Name 'Root Mean Square'?

The calculation process explains the name:

1. Square the signal values
2. Take the Mean (average) of the squared values
3. Take the square Root of that mean

This sequence ensures the result is always positive and properly weights large excursions (which contribute more power).

In practical communication systems, signal amplitudes span enormous ranges—from microvolt receiver inputs to kilovolt transmitter outputs. The decibel (dB) provides a logarithmic scale that compresses this range into manageable numbers.

Basic Definition (Power Ratio):

The decibel expresses power ratios:

Gain(dB) = 10 log₁₀(P₂/P₁)

For voltage ratios (assuming equal impedances):

Gain(dB) = 20 log₁₀(V₂/V₁)

The factor of 20 (instead of 10) arises because power is proportional to voltage squared.

Why Logarithmic?

1. Range Compression: A million-to-one power ratio becomes just 60 dB
2. Multiplicative → Additive: Cascaded gains multiply; in dB, they add. Three amplifiers of 10 dB each give 30 dB total.
3. Human Perception: Our senses (hearing, vision) respond logarithmically to stimuli. The dB scale matches subjective experience.

Important Reference Points:

• 0 dB = no change (P₂ = P₁)
• +3 dB ≈ double power
• +6 dB ≈ double voltage (quadruple power)
• +10 dB = 10× power
• +20 dB = 100× power
• −3 dB ≈ half power
• −10 dB = 1/10 power

Signal Level in Practice:

Consider a typical WiFi system:

• Transmitter output: +20 dBm (100 mW)
• Cable loss: −3 dB
• Antenna into cable: +17 dBm
• Antenna gain: +6 dBi
• EIRP (Effective Isotropic Radiated Power): +23 dBm
• Path loss at 50 meters: −70 dB
• Signal at receiver antenna: −47 dBm
• Receiver sensitivity: −80 dBm
• Margin: 33 dB (comfortable)

This 'link budget' analysis is fundamental to all wireless system design. Every component's contribution is simply added (in dB), making complex calculations straightforward.

Frequency measures how rapidly a signal oscillates—the number of complete cycles that occur in one second. While amplitude describes the magnitude of displacement, frequency describes the rate of that displacement.

Fundamental Definitions:

Frequency (f): The number of complete cycles per second, measured in Hertz (Hz).

• 1 Hz = 1 cycle per second
• 1 kHz = 1,000 Hz
• 1 MHz = 1,000,000 Hz
• 1 GHz = 1,000,000,000 Hz

Period (T): The time required for one complete cycle, measured in seconds.

• T = 1/f
• For f = 60 Hz: T = 16.67 milliseconds
• For f = 2.4 GHz: T = 0.417 nanoseconds

Angular Frequency (ω): Frequency expressed in radians per second.

• ω = 2πf radians/second
• One cycle = 2π radians
• Useful in mathematical analysis involving trigonometric functions

Wavelength (λ): The spatial length of one cycle in a propagating wave.

• λ = v/f (where v = propagation velocity)
• In free space: λ = c/f (c ≈ 3 × 10⁸ m/s)
• Higher frequencies → shorter wavelengths

The relationship between frequency and wavelength is fundamental to understanding wave propagation and antenna design. This relationship is governed by the propagation velocity of the medium:

λ = v/f or equivalently v = λf

Where:

• λ = wavelength (meters)
• v = propagation velocity (meters/second)
• f = frequency (Hertz)

In Free Space:

Electromagnetic waves travel at the speed of light: c = 299,792,458 m/s ≈ 3 × 10⁸ m/s

Therefore: λ = c/f

Practical Calculations:

For quick estimates in air/vacuum:

• λ (meters) = 300/f (MHz)
• λ (cm) = 30/f (GHz)

Examples:

• FM radio at 100 MHz: λ = 3 meters
• WiFi at 2.4 GHz: λ = 12.5 cm
• WiFi at 5 GHz: λ = 6 cm
• 5G mmWave at 28 GHz: λ = 1.07 cm
• Optical fiber at 193 THz: λ = 1.55 micrometers

Why Wavelength Matters:

1. Antenna Design: Efficient antennas are typically fractions of a wavelength (λ/4, λ/2). A 100 MHz antenna would be ~75 cm; a 1 GHz antenna would be ~7.5 cm.

2. Diffraction and Penetration: Longer wavelengths diffract around obstacles better. AM radio (λ ~ 300m) reaches behind buildings; mmWave 5G (λ ~ 1cm) is blocked by a hand.

3. Resolution: Imaging systems can resolve features no smaller than ~λ. Microwave imaging (cm resolution) vs. optical microscopy (μm resolution).

4. Atmospheric Effects: Different wavelengths interact differently with atmospheric gases, rain, and dust.

Bandwidth is the range of frequencies that a signal occupies or that a channel can accommodate. It is one of the most important parameters in communication system design, directly determining data capacity.

Signal Bandwidth:

Real signals are never perfectly sinusoidal—they contain energy across a range of frequencies. The bandwidth is typically defined as:

BW = fhigh - flow

Where fhigh and flow are the upper and lower frequency bounds containing significant signal energy.

Bandwidth Definitions:

3 dB Bandwidth: The frequency range where signal power remains within 3 dB (half) of the peak value. Most common definition.

Null-to-Null Bandwidth: The frequency range between the first zeros (nulls) of the signal spectrum. Common for digital signals.

Noise Equivalent Bandwidth: The bandwidth of an ideal rectangular filter that would pass the same noise power.

Occupied Bandwidth: Regulatory definition—the frequency range containing a specified percentage (usually 99%) of signal power.

Why Bandwidth Matters:

The Shannon-Hartley theorem establishes that channel capacity (maximum error-free data rate) is:

C = B log₂(1 + SNR) bits per second

Where B is bandwidth in Hz and SNR is the signal-to-noise ratio. This means:

• More bandwidth = more capacity (linear relationship)
• Higher SNR = more capacity (logarithmic relationship—diminishing returns)

Bandwidth is the scarce resource in wireless communications, which is why spectrum auctions generate billions of dollars.

Real-world signals rarely consist of a single pure frequency. Understanding the spectral composition of signals—their distribution of energy across frequencies—is essential for system design.

Harmonics:

When a nonlinear system processes a sinusoidal input, it generates harmonics—additional frequency components at integer multiples of the fundamental frequency:

• Fundamental: f₀
• 2nd harmonic: 2f₀
• 3rd harmonic: 3f₀
• nth harmonic: nf₀

Example: Square Wave Decomposition

A perfect square wave contains: f(t) = (4/π)[sin(ω₀t) + (1/3)sin(3ω₀t) + (1/5)sin(5ω₀t) + ...]

• Only odd harmonics (1st, 3rd, 5th, ...)
• Amplitudes decrease as 1/n
• Requires infinite harmonics for perfect square corners

Practically, this means square waves need significant bandwidth. A 1 MHz square wave needs harmonics to at least 5 MHz (5th harmonic) for reasonable fidelity, and higher for sharp edges.

Why This Matters for Digital Signals:

Digital signals are essentially modified square waves. A 1 Gbps data stream switches at up to 500 MHz fundamental frequency, requiring bandwidth to several GHz for clean signal integrity. This is why high-speed digital design requires RF thinking—multi-gigabit systems operate at microwave frequencies.

Spurious Emissions:

Nonlinear devices (amplifiers, mixers) generate unwanted harmonics that can interfere with other systems. Regulatory limits specify how much harmonic energy transmitters may emit. Filters are used to suppress harmonics before transmission.

The fundamental purpose of understanding amplitude and frequency is to enable modulation—the process of encoding information onto carrier signals. By systematically varying amplitude, frequency, or both, we transmit data through physical media.

Amplitude-Based Modulation:

Amplitude Modulation (AM): The carrier amplitude varies in proportion to the message signal while frequency remains constant.

s(t) = [Ac + Am cos(2πfm t)] cos(2πfc t)

Where Ac is carrier amplitude, Am is modulation depth, fc is carrier frequency, and fm is message frequency.

Amplitude Shift Keying (ASK): Digital variant—the carrier amplitude takes discrete levels to represent binary data.

• OOK (On-Off Keying): Amplitude switches between 0 and A
• Multi-level ASK: Multiple amplitude levels encode multiple bits per symbol

Frequency-Based Modulation:

Frequency Modulation (FM): The carrier frequency varies in proportion to the message signal while amplitude remains constant.

s(t) = Ac cos(2πfc t + 2πkf ∫m(t)dt)

Where kf is the frequency deviation constant.

Frequency Shift Keying (FSK): Digital variant—the carrier frequency shifts between discrete values.

• Binary FSK: Two frequencies represent 0 and 1
• Multi-level FSK: Multiple frequencies for more bits per symbol

Why Both Are Used:

AM is simpler but more susceptible to noise (noise adds directly to amplitude). FM is more robust but requires more bandwidth. The choice depends on constraints:

• AM: Simpler receivers, narrower bandwidth
• FM: Better noise immunity, constant envelope (easier amplification)

We have thoroughly explored amplitude and frequency—the two fundamental properties that characterize every analog signal and enable all modulation schemes. Let's consolidate the essential concepts:

What's Next:

We have covered two of the three fundamental signal properties. The next page explores Phase—the temporal position of a waveform relative to a reference. Phase is the key to understanding interference, coherent detection, and advanced modulation techniques like PSK and QAM that enable modern high-speed communications.