# Foreword to Noise in Analog Modulation Systems

Hello everyone, I hope that you all are doing good in your lives. As we have gone through features of amplitude and frequency modulation in the last blogs. We are now ready to focus on the effects of continuous wave modulation by increasing the effect of noise on the performance and thereby develop a better understanding to analog systems. However, to understand the analysis of noise in analog modulation systems, we need to do a number of things.

First and foremost we must have a receiver model. In the formulation of such a model basic practice is to model the receiver noise as white, additive, and Gaussian. However, these basic assumptions allow us to understand the way of reformation of the system under the increased effect of the noise signals.

Moreover, it provides a base for the comparison of the noise performance of different CW modulation schemes or simple types. Besides if you haven’t had a view of the last blogs; and because of which you are still unaware of the basics required for understanding today’s concepts; then please to refer the blogs mention below.

**Table of Contents**hide

### Receiver Model

The idea of modeling is fundamental to the study of all physical systems, including communication systems. **Through modeling, we improve our understanding of the capabilities and limitations of the system**. While formulating a receiver model for the study of noise in CW modulation system we need to keep these points into our mind;

- The model provides us with the enough information about the form of receiver noise that is present.
- Model accounts for the basic filtering and modulating features of the system.
- Lastly model is simple for statistical analysis of the system.

For such a situation; we propose to use the receiver model shown below in the figure below.

In the model above **s(t) denotes incoming modulated signal** and on the other hand, w(t) denotes front end receiver noise. Therefore the finally received signal is a sum of incoming signal and noise present at the input. The bandpass filter in the model represents the combines the filtering action of the tuned amplifier used in the actual receiver signal for amplifying the signal before demodulating. However, the bandwidth of the bandpass signal is just wide enough that the signal passes through it without any breaks. And lastly, the type of demodulator in the model depends on nature to modulation.

### Input Signal to Noise Ratio (SNR)_{I} in analog modulation

The input signal to noise ratio is define as the ratio of the average power of modulated signal s(t) to the **average power of filtered noise n(t).**

### Output Signal to Noise Ratio (SNR)_{O} in analog modulation

Output signal to noise ratio is define as a **ratio of the **average power of demodulated message to the average power of noise, both measure at the receiver output. It, however, provides an in-built measure for describing the fidelity with which the demodulation process in the receiver recovers the original signal from the modulated signal with the presence of noise.

The **output signal to noise ratio depends on other factors like**; types of modulation present in the transmitter and type of demodulator present in the receiver. Thus we can say it is informative to compare the output signal to noise ratios for different modulation and demodulation systems. However the comparison to be effective, it must be made on equal bases as describe below

- The modulated signal s(t) which is transmit by each system having the same average power.
- The front end noise receiver w(t) has the same average power measured at the message bandwidth W.

### Channel Signal to Noise Ratio (SNR)_{C}

According to a frame reference, channel signal to noise **r****atio is define as the ratio between the **average power of the modulated signal to the average power of noise in the message bandwidth, both of these are measure at the receiver input. These ratios maybe also view as a signal-to-noise ratio that results from baseband transmission of message signal m(t) without any kind of modulation.

### Figure Of Merit

For the purpose of comparing different continuous wave modulation systems. We normalize the receiver response by** dividing the output signal to noise ratio by the channel signal to noise ratio**. We thus define a figure of merit for the receiver as follows;

Clearly, the higher the value of the figure of merit, the better will the noise response of the receiver is. The figure of merit may equal one, be less than one, or greater than one, depending on the type of modulation used.

### Noise in DSB – SC Receiver

The noise analysis of a DSB – SC receiver using coherent detection is the simplest of all. The figure below shows the model of a DSB -SC receiver.

The use of such detection requires multiplication of the filtered signal x(t) by a locally generated sine wave cos(2πf_{c}t) and then low pass filtering the product. To simplify the analysis, we assume that the amplitude of the locally generated sine wave is unity. For this demodulation scheme to work nicely it is necessary that the local oscillator is in sync with both in-phase and frequency. However, we assume that both these are achieved.

The **DSB – SC component of filter signal x(t)** is express as;

where A Cos(2πf_{c}t) is the sinusoidal carrier wave and m(t) is the message signal. In the equation for s(t) we have the involvement of a system depending on scaling factor C. However the need for this is to make sure that the signal component s(t) is measure in the same unit as that of the noise component n(t). We assume m(t) as a sampling function of a stationary process of zero means. Whose power spectral density S_{M}(f) is limited to maximum frequency W. However W is the message bandwidth.

The **average power P of the message signal** is the area under the curve of power spectral density;

The carrier wave is analytically independent of the message signal. To focus on this independence, the carrier should have a random phase that is uniformly distributed over 2π radians. However, this phase is not mentioned while finding the equation of s(t) for convenience.

#### Channel Signal to Noise ratio in analog modulation

We may express the mean power of DSB – SC modulated signal component s(t) as C^{2}A_{C}^{2}P/2. With a noise spectral density of N_{0}/2; thus the mean of noise power in the message bandwidth W is equal to WN_{0}.

Thus the **channel signal to noise ratio of DSB – SC **modulation system is therefore expressed as;

Where the C^{2} in the numerator ensures that the ratio is without any unit.

#### Output signal to noise ratio in analog modulation

Lastly, the **output signal to noise ratio of DSB – SC** modulating system is;

#### Figure of merit of DSCB -SC

From both the above equation of channel signal to noise ratio and output signal to noise we get the** ****figure of merit as**;

**Note that **the factor C2 is common to both the channel and output signal to noise ratio and therefore cancels out each other in the evaluation process.

We should **also note that** the coherent detector output at the receiver using DSB – SC modulation; the translate signal sums coherently whereas, the translated noise band sun incoherently. This in turn means that the output signal to noise ratio in this receiver is twice the signal to noise ratio at coherent detector input.

### Conclusion

However here we are at the end of this particular part of the blog. Yes you all are right there are few more parts to these particular topics but we will not be able to cover them all in this particular part so you all will have to wait for the next part. In the next part, we are going to get to know about noise in AM receiver, Single tone modulation, threshold effect, Noise in FM receiver, Capture effect, and much more.

Besides, I hope that you all liked the content and have got all your answers. If you like blogs then please make to share them with others and also don’t forget to comment on the part which you liked the most. And also feel free to comment down and ask any doubts if there. And please do suggest the topic which you will like to read next on.

Have a nice day

Regards 🙂

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