Can Phase and Amplitude be Modified for Transmission Purposes?

In the world of telecommunications, signal processing, and vibration measurement, the ability to modify the phase and amplitude of a signal is crucial for effective data transmission and analysis. This comprehensive guide delves into the intricacies of how phase and amplitude can be manipulated for various transmission purposes, providing a technical and advanced understanding for electronics students and enthusiasts.

Modifying Phase and Amplitude for Telecommunications

In telecommunications, the transmission function involves the physical lines, radio waves, data communication channels, and wiring that transmit data from one point to another. This transmission function can be modified by adjusting the phase and amplitude of the signal being transmitted. Several techniques are employed to achieve this:

1. Amplitude Modulation (AM): The formula for AM is:
   s(t) = Acos(2πft) + (B/2)cos(2π(f+fm)t) + (B/2)cos(2π(f-fm)t)
   where s(t) is the modulated signal, Acos(2πft) is the carrier signal, B is the amplitude of the modulating signal, fm is the frequency of the modulating signal, and f is the frequency of the carrier signal.

2. Phase Modulation (PM): The formula for PM is:
   s(t) = Acos(2πft + Φm)
   where s(t) is the modulated signal, Acos(2πft) is the carrier signal, Φm is the phase shift of the modulating signal, and f is the frequency of the carrier signal.

3. Quadrature Amplitude Modulation (QAM): The formula for QAM is:
   s(t) = Icos(2πft) + Qsin(2πft)
   where s(t) is the modulated signal, I and Q are the in-phase and quadrature components of the modulating signal, and f is the frequency of the carrier signal.

These techniques allow for the modification of both the phase and amplitude of the signal, enabling efficient data transmission and encoding.

Modifying Phase and Amplitude for Vibration Measurement

In the field of vibration measurement, the amplitude of a vibration can be quantified in several ways, including peak-to-peak level, peak level, average level, and RMS level. The phase and amplitude of a vibration can be modified using various techniques, such as vibration isolation or reduction, which can involve changing the physical properties of the system being measured or using electronic devices to modify the signal.

Table 1: Vibration Amplitude Quantification

Quantification Method	Symbol	Formula
Peak-to-Peak Level	APP	APP = 2A
Peak Level	AP	AP = A
Average Level	AA	AA = (1/T)∫
RMS Level	ARMS	ARMS = sqrt[(1/T)∫x^2(t)dt]

where A is the amplitude of the vibration, T is the period of the vibration, and x(t) is the vibration signal.

Modifying Phase and Amplitude for Quantitative Phase Imaging (QPI)

Quantitative phase imaging (QPI) is a technique that allows for the reconstruction of the 3D refractive index of an object by measuring the complex amplitude distribution of the scattered fields. In transmission QPI, light traverses the entire sample, accumulating phase shifts relative to the reference wave. By varying the angle of illumination, the 3D scattering potential of an object can be reconstructed with high resolution, which depends on the accessible range in the 3D Fourier space.

In reflection QPI modalities, strong backscattered light from internal interfaces can enable the extraction of quantitative phase information at these interfaces, providing depth selectivity. Depending on the location of the coherence gate, the quantitative phase shift between the reference and sample wave can be measured either at the sample surface or within subsurface tissue layers with strong refractive index mismatches.

Examples and Numerical Problems

Example 1: Amplitude Modulation (AM)
Suppose we have a carrier signal with a frequency of 1 MHz and an amplitude of 1 V. We want to modulate this signal with a sinusoidal signal with a frequency of 10 kHz and an amplitude of 0.5 V. The modulated signal will be:

s(t) = 1cos(2π10^6t) + (0.5/2)cos(2π(10^6+10^4)t) + (0.5/2)cos(2π(10^6-10^4)t)

Example 2: Phase Modulation (PM)
Suppose we have a carrier signal with a frequency of 1 MHz and a phase of 0 degrees. We want to modulate this signal with a sinusoidal signal with a frequency of 10 kHz and a phase of 90 degrees. The modulated signal will be:

s(t) = 1cos(2π10^6t + 90 degrees)

Example 3: Quadrature Amplitude Modulation (QAM)
Suppose we have a carrier signal with a frequency of 1 MHz. We want to modulate this signal with two sinusoidal signals, one with a frequency of 10 kHz and an amplitude of 0.5 V, and the other with a frequency of 20 kHz and an amplitude of 0.3 V. The modulated signal will be:

s(t) = 0.5cos(2π10^6t) + 0.3sin(2π10^6t)

Numerical Problem 1: Amplitude Modulation (AM)
Suppose we have a carrier signal with a frequency of 1 MHz and an amplitude of 1 V. We want to modulate this signal with a sinusoidal signal with a frequency of 10 kHz and an amplitude of 0.5 V using AM. What is the modulated signal?

Numerical Problem 2: Phase Modulation (PM)
Suppose we have a carrier signal with a frequency of 1 MHz and a phase of 0 degrees. We want to modulate this signal with a sinusoidal signal with a frequency of 10 kHz and a phase of 90 degrees using PM. What is the modulated signal?

Numerical Problem 3: Quadrature Amplitude Modulation (QAM)
Suppose we have a carrier signal with a frequency of 1 MHz. We want to modulate this signal with two sinusoidal signals, one with a frequency of 10 kHz and an amplitude of 0.5 V, and the other with a frequency of 20 kHz and an amplitude of 0.3 V using QAM. What is the modulated signal?

In conclusion, the ability to modify the phase and amplitude of a signal is crucial in various fields, including telecommunications, signal processing, and vibration measurement. The techniques discussed in this guide, such as AM, PM, QAM, vibration isolation or reduction, and QPI, provide a comprehensive understanding of how these modifications can be achieved and applied in practical scenarios.

References:
1. Transmission Function – an overview | ScienceDirect Topics
2. Quantitative phase‐amplitude microscopy II: differential interference … – Wiley Online Library
3. Vibration Measurement: The Complete Guide – HBK