COMPS-a)A carrier signal of frequency 99 MHz is fr

Sure, let's break down each part of the question and provide the necessary answers. Part a) A carrier signal of frequency 99 MHz is frequency-modulated by a message signal of bandwidth 5 kHz such that the largest frequency deviation is 0.1 MHz. Determine whether this is narrow-band FM (NBFM) or wide-band FM (WBFM) and estimate the bandpass bandwidth. To determine whether the FM is narrow-band or wide-band, we use the modulation index

\beta

:

\beta = \frac{\Delta f}{f_m}

where

\Delta f

is the frequency deviation and

f_m

is the message signal bandwidth. Given: -

\Delta f = 0.1 \text{ MHz} = 100 \text{ kHz}

-

f_m = 5 \text{ kHz}

So,

\beta = \frac{100 \text{ kHz}}{5 \text{ kHz}} = 20

Since \beta > 1, this is wide-band FM (WBFM). The bandpass bandwidth for WBFM can be estimated using Carson's Rule:

BW = 2(\Delta f + f_m)

BW = 2(100 \text{ kHz} + 5 \text{ kHz}) = 210 \text{ kHz}

A

Key Concept

Wide-band FM (WBFM) determination

Explanation

The modulation index

\beta

is greater than 1, indicating WBFM. The bandpass bandwidth is estimated using Carson's Rule.

Part b) Show that the NBFM expression for a single tone message can be written as a scaled version of where

W_m

and

W_C

are the (single-tone) message and carrier frequencies, respectively, and is the modulation index. What is the bandpass bandwidth of the above? For NBFM, the modulation index

\beta

is much less than 1. The NBFM signal can be approximated as:

s(t) \approx A_c \cos(2\pi f_c t) - A_c \beta \sin(2\pi f_c t) \sin(2\pi f_m t)

Using trigonometric identities, this can be written as:

s(t) \approx A_c \cos(2\pi f_c t) - \frac{A_c \beta}{2} \left[ \cos(2\pi (f_c - f_m) t) - \cos(2\pi (f_c + f_m) t) \right]

The bandpass bandwidth for NBFM is approximately:

BW \approx 2 f_m

B

Key Concept

NBFM signal expression and bandwidth

Explanation

The NBFM signal can be approximated using trigonometric identities, and its bandwidth is approximately twice the message frequency.

Part c) Draw the block diagram of a coherent DSB detector and discuss whether this could be used to recover the baseband message from a NBFM signal. Use appropriate diagrams, signal sketches, and mathematical steps to support and illustrate your answer. A coherent DSB (Double Sideband) detector typically consists of a product modulator followed by a low-pass filter. Here is a simple block diagram:

To recover the baseband message from an NBFM signal, the coherent DSB detector is not suitable because it is designed for amplitude modulation (AM) signals. NBFM signals require a frequency discriminator or a phase-locked loop (PLL) for proper demodulation.

markdown | Feature | Frequency Modulation (FM) | Amplitude Modulation (AM) | Double Sideband (DSB) | Single Sideband (SSB) | |--------------------|---------------------------|---------------------------|-----------------------|-----------------------| | Power Efficiency | High | Low | Moderate | High | | Bandpass Bandwidth | Wide (WBFM) / Narrow (NBFM)| Narrow | Wide | Narrow | | Complexity | Moderate | Low | Moderate | High | | Example Applications | Radio Broadcasting, TV | AM Radio | TV Audio | Long-distance Communication |

*****D***

The modulation format depicted in the spectrum analyzer screenshot is Double Sideband-Suppressed Carrier (DSB-SC) modulation.

The regions of the spectrum highlighted and labeled X and Y represent the upper sideband (USB) and lower sideband (LSB) respectively.

These sidebands originate from the modulation process where the carrier signal is modulated by the message signal, resulting in the creation of two sidebands that carry the same information.

The presence of these sidebands is crucial for the recovery of the baseband message signal. If either sideband is missing or distorted, it can affect the quality and integrity of the recovered signal.

The carrier frequency is 100.000 MHz.

The bandpass bandwidth is the span of frequencies that the modulated signal occupies, which is 1.000 MHz.

The baseband message bandwidth is half of the bandpass bandwidth, which is 0.500 MHz.

The modulation index can be expressed as:

\text{Modulation Index} = \frac{E_{LSB} + E_{USB}}{E_C}

Given that

E_{LSB} = 18 \text{ mV}

,

E_{USB} = 20 \text{ mV}

, and

E_C = 76 \text{ mV}

, the percentage of the bandpass signal power contained in the lower sideband can be calculated as follows:

\text{Percentage of Power in LSB} = \left( \frac{E_{LSB}}{E_{LSB} + E_{USB} + E_C} \right) \times 100

Substituting the given values:

\text{Percentage of Power in LSB} = \left( \frac{18}{18 + 20 + 76} \right) \times 100 \approx 15.52\%

The signal is under-modulated because the modulation index is less than 1.

please answer these five quertions

Sure, let's address each of the five questions based on the provided spectrum analyzer screenshot and the detailed description. a) Name the modulation format and describe the regions of the spectrum highlighted and labelled X and Y. Discuss where these may have originated and how, if at all, they might affect the recovered signal. The modulation format is Amplitude Modulation (AM). Regions X and Y are likely to be the lower and upper sidebands, respectively. These sidebands originate from the modulation process where the baseband message signal modulates the carrier frequency. They contain the information of the original message signal. Any interference or noise in these regions can affect the quality of the recovered signal.

A

Key Concept

Amplitude Modulation (AM)

Explanation

In AM, the carrier frequency is modulated by the baseband message signal, creating sidebands that contain the message information.

b) Find the carrier frequency and bandpass and baseband message bandwidths. - Carrier frequency (

f_c

): 100.000 MHz - Bandpass message bandwidth: The span of the spectrum is 1.000 MHz, so the bandpass bandwidth is 1.000 MHz. - Baseband message bandwidth: Since the bandpass bandwidth is twice the baseband bandwidth, the baseband bandwidth is 0.500 MHz.

B

Key Concept

Carrier and Bandwidth Calculation

Explanation

The carrier frequency is the central frequency, and the bandpass bandwidth is the total span of the spectrum. The baseband bandwidth is half of the bandpass bandwidth.

c) Starting from the mathematical representation of this modulation format, show that the modulation index can be expressed as

\frac{E_{LSB} + E_{USB}}{E_c}

. The modulation index (

m

) in AM can be defined as the ratio of the peak amplitude of the message signal to the peak amplitude of the carrier signal. Mathematically, it can be shown as:

m = \frac{E_{LSB} + E_{USB}}{E_c}

Where: -

E_{LSB}

is the amplitude of the lower sideband. -

E_{USB}

is the amplitude of the upper sideband. -

E_c

is the amplitude of the carrier.

C

Key Concept

Modulation Index in AM

Explanation

The modulation index is a measure of the extent of modulation applied to the carrier signal, calculated using the amplitudes of the sidebands and the carrier.

d) Find the percentage of the bandpass signal power contained in the lower sideband, given that

E_{LSB} = 18

mV,

E_{USB} = 20

mV, and

E_c = 76

mV. The power in the lower sideband (

P_{LSB}

) can be calculated using the formula:

P_{LSB} = \left(\frac{E_{LSB}}{E_c}\right)^2 \times 100\%

Substituting the given values:

P_{LSB} = \left(\frac{18}{76}\right)^2 \times 100\% \approx 5.61\%

D

Key Concept

Power Calculation in Sidebands

Explanation

The power in a sideband is proportional to the square of the ratio of the sideband amplitude to the carrier amplitude.

e) Is this signal over-modulated, under-modulated or perfectly-modulated? Explain your answer. To determine if the signal is over-modulated, under-modulated, or perfectly-modulated, we need to calculate the modulation index (

m

):

m = \frac{E_{LSB} + E_{USB}}{E_c} = \frac{18 + 20}{76} \approx 0.5

Since the modulation index is less than 1, the signal is under-modulated.

E

Key Concept

Modulation Index Interpretation

Explanation

A modulation index less than 1 indicates under-modulation, equal to 1 indicates perfect modulation, and greater than 1 indicates over-modulation.

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