Discuss Analog Communication The communication based on analog signals and analog values is known as Analog Communication. This tutorial provides knowledge on the various modulation techniques that are useful in Analog Communication systems. By the completion of this tutorial, the reader will be able to understand the conceptual details involved in analog communication. Learning working make money
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Analog Communication – Quick Guide Analog Communication – Introduction The word communication arises from the Latin word commūnicāre, which means “to share”. Communication is the basic step for exchange of information. For example, a baby in a cradle, communicates with a cry when she needs her mother. A cow moos loudly when it is in danger. A person communicates with the help of a language. Communication is the bridge to share. Communication can be defined as the process of exchange of information through means such as words, actions, signs, etc., between two or more individuals. Parts of a Communication System Any system, which provides communication consists of the three important and basic parts as shown in the following figure. Sender is the person who sends a message. It could be a transmitting station from where the signal is transmitted. Channel is the medium through which the message signals travel to reach the destination. Receiver is the person who receives the message. It could be a receiving station where the transmitted signal is being received. Types of Signals Conveying an information by some means such as gestures, sounds, actions, etc., can be termed as signaling. Hence, a signal can be a source of energy which transmits some information. This signal helps to establish a communication between the sender and the receiver. An electrical impulse or an electromagnetic wave which travels a distance to convey a message, can be termed as a signal in communication systems. Depending on their characteristics, signals are mainly classified into two types: Analog and Digital. Analog and Digital signals are further classified, as shown in the following figure. Analog Signal A continuous time varying signal, which represents a time varying quantity can be termed as an Analog Signal. This signal keeps on varying with respect to time, according to the instantaneous values of the quantity, which represents it. Example Let us consider a tap that fills a tank of 100 liters capacity in an hour (6 AM to 7 AM). The portion of filling the tank is varied by the varying time. Which means, after 15 minutes (6:15 AM) the quarter portion of the tank gets filled, whereas at 6:45 AM, 3/4th of the tank is filled. If we try to plot the varying portions of water in the tank according to the varying time, it would look like the following figure. As the result shown in this image varies (increases) according to time, this time varying quantity can be understood as Analog quantity. The signal which represents this condition with an inclined line in the figure, is an Analog Signal. The communication based on analog signals and analog values is called as Analog Communication. Digital Signal A signal which is discrete in nature or which is non-continuous in form can be termed as a Digital signal. This signal has individual values, denoted separately, which are not based on the previous values, as if they are derived at that particular instant of time. Example Let us consider a classroom having 20 students. If their attendance in a week is plotted, it would look like the following figure. In this figure, the values are stated separately. For instance, the attendance of the class on Wednesday is 20 whereas on Saturday is 15. These values can be considered individually and separately or discretely, hence they are called as discrete values. The binary digits which has only 1s and 0s are mostly termed as digital values. Hence, the signals which represent 1s and 0s are also called as digital signals. The communication based on digital signals and digital values is called as Digital Communication. Periodic Signal Any analog or digital signal, that repeats its pattern over a period of time, is called as a Periodic Signal. This signal has its pattern continued repeatedly and is easy to be assumed or to be calculated. Example If we consider a machinery in an industry, the process that takes place one after the other is a continuous procedure. For example, procuring and grading the raw material, processing the material in batches, packing a load of products one after the other, etc., follows a certain procedure repeatedly. Such a process whether considered analog or digital, can be graphically represented as follows. Aperiodic Signal Any analog or digital signal, that doesn’t repeat its pattern over a period of time is called as Aperiodic Signal. This signal has its pattern continued but the pattern is not repeated. It is also not so easy to be assumed or to be calculated. Example The daily routine of a person, if considered, consists of various types of work which take different time intervals for different tasks. The time interval or the work doesn’t continuously repeat. For example, a person will not continuously brush his teeth from morning to night, that too with the same time period. Such a process whether considered analog or digital, can be graphically represented as follows. In general, the signals which are used in communication systems are analog in nature, which are transmitted in analog or converted to digital and then transmitted, depending upon the requirement. Analog Communication – Modulation For a signal to be transmitted to a distance, without the effect of any external interferences or noise addition and without getting faded away, it has to undergo a process called as Modulation. It improves the strength of the signal without disturbing the parameters of the original signal. What is Modulation? A message carrying a signal has to get transmitted over a distance and for it to establish a reliable communication, it needs to take the help of a high frequency signal which should not affect the original characteristics of the message signal. The characteristics of the message signal, if changed, the message contained in it also alters. Hence, it is a must to take care of the message signal. A high frequency signal can travel up to a longer distance, without getting affected by external disturbances. We take the help of such high
Analog Communication – Transmitters The antenna present at the end of transmitter section, transmits the modulated wave. In this chapter, let us discuss about AM and FM transmitters. AM Transmitter AM transmitter takes the audio signal as an input and delivers amplitude modulated wave to the antenna as an output to be transmitted. The block diagram of AM transmitter is shown in the following figure. The working of AM transmitter can be explained as follows. The audio signal from the output of the microphone is sent to the pre-amplifier, which boosts the level of the modulating signal. The RF oscillator generates the carrier signal. Both the modulating and the carrier signal is sent to AM modulator. Power amplifier is used to increase the power levels of AM wave. This wave is finally passed to the antenna to be transmitted. FM Transmitter FM transmitter is the whole unit, which takes the audio signal as an input and delivers FM wave to the antenna as an output to be transmitted. The block diagram of FM transmitter is shown in the following figure. The working of FM transmitter can be explained as follows. The audio signal from the output of the microphone is sent to the pre-amplifier, which boosts the level of the modulating signal. This signal is then passed to high pass filter, which acts as a pre-emphasis network to filter out the noise and improve the signal to noise ratio. This signal is further passed to the FM modulator circuit. The oscillator circuit generates a high frequency carrier, which is sent to the modulator along with the modulating signal. Several stages of frequency multiplier are used to increase the operating frequency. Even then, the power of the signal is not enough to transmit. Hence, a RF power amplifier is used at the end to increase the power of the modulated signal. This FM modulated output is finally passed to the antenna to be transmitted. Learning working make money
Analog Communication – FM Demodulators In this chapter, let us discuss about the demodulators which demodulate the FM wave. The following two methods demodulate FM wave. Frequency discrimination method Phase discrimination method Frequency Discrimination Method We know that the equation of FM wave is $$sleft ( t right ) =A_c cos left ( 2 pi f_ct+2 pi k_f int mleft ( t right )dt right )$$ Differentiate the above equation with respect to ”t”. $$frac{dsleft ( t right )}{dt}= -A_cleft ( 2 pi f_c+2 pi k_fmleft ( t right ) right ) sinleft ( 2 pi f_ct+2 pi k_fint mleft ( t right )dt right )$$ We can write, $-sin theta$ as $sin left ( theta -180^0 right )$. $$Rightarrow frac{ds(t)}{dt}=A_cleft ( 2 pi f_c+2 pi k_fmleft ( t right ) right )sinleft ( 2 pi f_ct+2 pi k_f int mleft ( t right )dt-180^0 right )$$ $$Rightarrow frac{ds(t)}{dt}=A_cleft ( 2 pi f_c right )left [ 1+left ( frac{k_f}{k_c} right )mleft ( t right ) right ] sinleft ( 2 pi f_ct+2 pi k_fint mleft ( t right )dt-180^0 right )$$ In the above equation, the amplitude term resembles the envelope of AM wave and the angle term resembles the angle of FM wave. Here, our requirement is the modulating signal $mleft ( t right )$. Hence, we can recover it from the envelope of AM wave. The following figure shows the block diagram of FM demodulator using frequency discrimination method. This block diagram consists of the differentiator and the envelope detector. Differentiator is used to convert the FM wave into a combination of AM wave and FM wave. This means, it converts the frequency variations of FM wave into the corresponding voltage (amplitude) variations of AM wave. We know the operation of the envelope detector. It produces the demodulated output of AM wave, which is nothing but the modulating signal. Phase Discrimination Method The following figure shows the block diagram of FM demodulator using phase discrimination method. This block diagram consists of the multiplier, the low pass filter, and the Voltage Controlled Oscillator (VCO). VCO produces an output signal $v left ( t right )$, whose frequency is proportional to the input signal voltage $d left ( t right )$. Initially, when the signal $d left ( t right )$ is zero, adjust the VCO to produce an output signal $v left ( t right )$, having a carrier frequency and $-90^0$ phase shift with respect to the carrier signal. FM wave $s left ( t right )$ and the VCO output $v left ( t right )$ are applied as inputs of the multiplier. The multiplier produces an output, having a high frequency component and a low frequency component. Low pass filter eliminates the high frequency component and produces only the low frequency component as its output. This low frequency component contains only the term-related phase difference. Hence, we get the modulating signal $m left ( t right )$ from this output of the low pass filter. Learning working make money
Analog Communication – Pulse Modulation After continuous wave modulation, the next division is Pulse modulation. In this chapter, let us discuss the following analog pulse modulation techniques. Pulse Amplitude Modulation Pulse Width Modulation Pulse Position Modulation Pulse Amplitude Modulation In Pulse Amplitude Modulation (PAM) technique, the amplitude of the pulse carrier varies, which is proportional to the instantaneous amplitude of the message signal. The pulse amplitude modulated signal will follow the amplitude of the original signal, as the signal traces out the path of the whole wave. In natural PAM, a signal sampled at Nyquist rate can be reconstructed, by passing it through an efficient Low Pass Filter (LPF) with exact cutoff frequency. The following figures explain the Pulse Amplitude Modulation. Though the PAM signal is passed through a LPF, it cannot recover the signal without distortion. Hence, to avoid this noise, use flat-top sampling. The flat-top PAM signal is shown in the following figure. Flat-top sampling is the process in which, the sampled signal can be represented in pulses for which the amplitude of the signal cannot be changed with respect to the analog signal, to be sampled. The tops of amplitude remain flat. This process simplifies the circuit design. Pulse Width Modulation In Pulse Width Modulation (PWM) or Pulse Duration Modulation (PDM) or Pulse Time Modulation (PTM) technique, the width or the duration or the time of the pulse carrier varies, which is proportional to the instantaneous amplitude of the message signal. The width of the pulse varies in this method, but the amplitude of the signal remains constant. Amplitude limiters are used to make the amplitude of the signal constant. These circuits clip off the amplitude to a desired level, and hence the noise is limited. The following figure explains the types of Pulse Width Modulations. There are three types of PWM. The leading edge of the pulse being constant, the trailing edge varies according to the message signal. The waveform for this type of PWM is denoted as (a) in the above figure. The trailing edge of the pulse being constant, the leading edge varies according to the message signal. The waveform for this type of PWM is denoted as (b) in the above figure. The center of the pulse being constant, the leading edge and the trailing edge varies according to the message signal. The waveform for this type of PWM is denoted as (c) shown in the above figure. Pulse Position Modulation Pulse Position Modulation (PPM) is an analog modulation scheme in which, the amplitude and the width of the pulses are kept constant, while the position of each pulse, with reference to the position of a reference pulse varies according to the instantaneous sampled value of the message signal. The transmitter has to send synchronizing pulses (or simply sync pulses) to keep the transmitter and the receiver in sync. These sync pulses help to maintain the position of the pulses. The following figures explain the Pulse Position Modulation. Pulse position modulation is done in accordance with the pulse width modulated signal. Each trailing edge of the pulse width modulated signal becomes the starting point for pulses in PPM signal. Hence, the position of these pulses is proportional to the width of the PWM pulses. Advantage As the amplitude and the width are constant, the power handled is also constant. Disadvantage The synchronization between the transmitter and the receiver is a must. Comparison between PAM, PWM, and PPM The following table presents the comparison between three modulation techniques. PAM PWM PPM Amplitude is varied Width is varied Position is varied Bandwidth depends on the width of the pulse Bandwidth depends on the rise time of the pulse Bandwidth depends on the rise time of the pulse Instantaneous transmitter power varies with the amplitude of the pulses Instantaneous transmitter power varies with the amplitude and the width of the pulses Instantaneous transmitter power remains constant with the width of the pulses System complexity is high System complexity is low System complexity is low Noise interference is high Noise interference is low Noise interference is low It is similar to amplitude modulation It is similar to frequency modulation It is similar to phase modulation Learning working make money
Analog Communication – Transducers Transducer is a device, which converts energy from one form to other. In this chapter, let us discuss about the transducers used in communication systems. Why do We Need Transducers? In the real world, communication between any two nearby persons takes place with the help of sound waves. But, if the persons are far away, then it is difficult to convey the information without any loss by using sound waves in its physical form. To overcome this difficulty, we can use modulators in the transmitter section and demodulators in the receiver section. These modulators and demodulators operate with electrical signals. That’s why we require a device, which has to convert the sound waves into electrical signals or vice versa. That device is known as a transducer. Following is a simple block diagram of a transducer. This transducer has a single input and a single output. It converts the energy present at the input into its equivalent output having another energy. Basically, a transducer converts the non-electrical form of energy into an electrical form or vice versa. Types of Transducers We can classify the transducers into following two types based on the placement (position) of the transducer in communication systems. Input Transducer Output Transducer Input Transducers The transducer present at the input of the communication system is known as an input transducer. Following is the block diagram of an input transducer. This input transducer converts the non-electrical physical quantity into an electrical signal. The physical quantities such as sound or light can be converted into electrical quantities such as voltage or current by using this transducer. Example: Microphone. Microphone is used as the input transducer, which is placed between the information source and the transmitter section. The information source produces the information in the form of sound waves. The microphone converts these sound waves into electrical signals with the help of a diaphragm. These electrical signals can be used for further processing. Output Transducers The transducer present at the output of communication system is known as output transducer. Following is the block diagram of an output transducer. This output transducer converts the electrical signal into non-electrical physical quantity. The electrical quantities such as voltage or current can be converted into physical quantities such as sound or light by using this transducer. Example: Loudspeaker. The loud speaker is used as the output transducer, which is placed between the receiver section and the destination. The demodulator present in the receiver section produces the demodulated output. So, the loud speaker converts the electrical signals (demodulated output) into sound waves. Therefore, the functionality of the loud speaker is just opposite to the functionality of the microphone. In addition to the above transducers, there is one more transducer which is used in communication systems. This transducer can be placed either at the end of the transmitter section or at the starting of the receiver section. Example: Antenna. An Antenna is a transducer, which converts electrical signals into electromagnetic waves and vice versa. An Antenna can be used either as a transmitting antenna or as a receiving antenna. A transmitting antenna converts electrical signals into electromagnetic waves and radiates them. While, a receiving antenna converts electromagnetic waves from the received beam into electrical signals. In this two-way communication, the same antenna can be used for both transmission and reception. Learning working make money
Analog Communication – Sampling So far, we have discussed about continuous-wave modulation. We will discuss about pulse modulation in the next chapter. These pulse modulation techniques deal with discrete signals. So, now let us see how to convert a continuous time signal into a discrete one. The process of converting continuous time signals into equivalent discrete time signals, can be termed as Sampling. A certain instant of data is continually sampled in the sampling process. The following figure shows a continuous-time signal x(t) and the corresponding sampled signal xs(t). When x(t) is multiplied by a periodic impulse train, the sampled signal xs(t) is obtained. A sampling signal is a periodic train of pulses, having unit amplitude, sampled at equal intervals of time $T_s$, which is called as sampling time. This data is transmitted at the time instants $T_s$ and the carrier signal is transmitted at the remaining time. Sampling Rate To discretize the signals, the gap between the samples should be fixed. That gap can be termed as the sampling period $T_s$. Reciprocal of the sampling period is known as sampling frequency or sampling rate $f_s$. Mathematically, we can write it as $$f_s= frac{1}{T_s}$$ Where, $f_s$ is the sampling frequency or the sampling rate $T_s$ is the sampling period Sampling Theorem The sampling rate should be such that the data in the message signal should neither be lost nor it should get over-lapped. The sampling theorem states that, “a signal can be exactly reproduced if it is sampled at the rate $f_s$, which is greater than or equal to twice the maximum frequency of the given signal W.” Mathematically, we can write it as $$f_sgeq 2W$$ Where, $f_s$ is the sampling rate $W$ is the highest frequency of the given signal If the sampling rate is equal to twice the maximum frequency of the given signal W, then it is called as Nyquist rate. The sampling theorem, which is also called as Nyquist theorem, delivers the theory of sufficient sample rate in terms of bandwidth for the class of functions that are bandlimited. For continuous-time signal x(t), which is band-limited in the frequency domain is represented as shown in the following figure. If the signal is sampled above Nyquist rate, then the original signal can be recovered. The following figure explains a signal, if sampled at a higher rate than 2w in the frequency domain. If the same signal is sampled at a rate less than 2w, then the sampled signal would look like the following figure. We can observe from the above pattern that there is over-lapping of information, which leads to mixing up and loss of information. This unwanted phenomenon of over-lapping is called as Aliasing. Aliasing can be referred to as “the phenomenon of a high-frequency component in the spectrum of a signal, taking on the identity of a low-frequency component in the spectrum of its sampled version.” Hence, the sampling rate of the signal is chosen to be as Nyquist rate. If the sampling rate is equal to twice the highest frequency of the given signal W, then the sampled signal would look like the following figure. In this case, the signal can be recovered without any loss. Hence, this is a good sampling rate. Learning working make money
Analog Communication – Receivers The antenna present at the beginning of the receiver section, receives the modulated wave. First let us discuss the requirements of a receiver. Requirements of a Receiver AM receiver receives AM wave and demodulates it by using the envelope detector. Similarly, FM receiver receives FM wave and demodulates it by using the Frequency Discrimination method. Following are the requirements of both AM and FM receiver. It should be cost-effective. It should receive the corresponding modulated waves. The receiver should be able to tune and amplify the desired station. It should have an ability to reject the unwanted stations. Demodulation has to be done to all the station signals, irrespective of the carrier signal frequency. For these requirements to be fulfilled, the tuner circuit and the mixer circuit should be very effective. The procedure of RF mixing is an interesting phenomenon. RF Mixing The RF mixing unit develops an Intermediate Frequency (IF) to which any received signal is converted, so as to process the signal effectively. RF Mixer is an important stage in the receiver. Two signals of different frequencies are taken where one signal level affects the level of the other signal, to produce the resultant mixed output. The input signals and the resultant mixer output is illustrated in the following figures. Let the first and second signal frequencies be $f_1$ and $f_2$. If these two signals are applied as inputs of RF mixer, then it produces an output signal, having frequencies of $f_1+f_2$ and $f_1-f_2$. If this is observed in the frequency domain, the pattern looks like the following figure. In this case, $f_1$ is greater than $f_2$. So, the resultant output has frequencies $f_1+f_2$ and $f_1-f_2$. Similarly, if $f_2$ is greater than $f_1$, then the resultant output will have the frequencies $f_1+f_2$ and $f_1-f_2$. AM Receiver The AM super heterodyne receiver takes the amplitude modulated wave as an input and produces the original audio signal as an output. Selectivity is the ability of selecting a particular signal, while rejecting the others. Sensitivity is the capacity of detecting RF signal and demodulating it, while at the lowest power level. Radio amateurs are the initial radio receivers. However, they have drawbacks such as poor sensitivity and selectivity. To overcome these drawbacks, super heterodyne receiver was invented. The block diagram of AM receiver is shown in the following figure. RF Tuner Section The amplitude modulated wave received by the antenna is first passed to the tuner circuit through a transformer. The tuner circuit is nothing but a LC circuit, which is also called as resonant or tank circuit. It selects the frequency, desired by the AM receiver. It also tunes the local oscillator and the RF filter at the same time. RF Mixer The signal from the tuner output is sent to the RF-IF converter, which acts as a mixer. It has a local oscillator, which produces a constant frequency. The mixing process is done here, having the received signal as one input and the local oscillator frequency as the other input. The resultant output is a mixture of two frequencies $left [ left ( f_1+f_2 right ) , left ( f_1-f_2 right )right ]$ produced by the mixer, which is called as the Intermediate Frequency (IF). The production of IF helps in the demodulation of any station signal having any carrier frequency. Hence, all signals are translated to a fixed carrier frequency for adequate selectivity. IF Filter Intermediate frequency filter is a band pass filter, which passes the desired frequency. It eliminates all other unwanted frequency components present in it. This is the advantage of IF filter, which allows only IF frequency. AM Demodulator The received AM wave is now demodulated using AM demodulator. This demodulator uses the envelope detection process to receive the modulating signal. Audio Amplifier This is the power amplifier stage, which is used to amplify the detected audio signal. The processed signal is strengthened to be effective. This signal is passed on to the loudspeaker to get the original sound signal. FM Receiver The block diagram of FM receiver is shown in the following figure. This block diagram of FM receiver is similar to the block diagram of AM receiver. The two blocks Amplitude limiter and De-emphasis network are included before and after FM demodulator. The operation of the remaining blocks is the same as that of AM receiver. We know that in FM modulation, the amplitude of FM wave remains constant. However, if some noise is added with FM wave in the channel, due to that the amplitude of FM wave may vary. Thus, with the help of amplitude limiter we can maintain the amplitude of FM wave as constant by removing the unwanted peaks of the noise signal. In FM transmitter, we have seen the pre-emphasis network (High pass filter), which is present before FM modulator. This is used to improve the SNR of high frequency audio signal. The reverse process of pre-emphasis is known as de-emphasis. Thus, in this FM receiver, the de-emphasis network (Low pass filter) is included after FM demodulator. This signal is passed to the audio amplifier to increase the power level. Finally, we get the original sound signal from the loudspeaker. Learning working make money
Analog Communication – SNR Calculations In this chapter, let us calculate Signal to Noise Ratios and Figure of Merits of various modulated waves, which are demodulated at the receiver. Signal to Noise Ratio Signal-to-Noise Ratio (SNR) is the ratio of the signal power to noise power. The higher the value of SNR, the greater will be the quality of the received output. Signal-to-Noise Ratio at different points can be calculated using the following formulas. Input SNR = $left ( SNR right )_I= frac{Average :: power ::of ::modulating ::signal}{Average:: power ::of ::noise ::at ::input}$ Output SNR = $left ( SNR right )_O= frac{Average :: power ::of ::demodulated ::signal}{Average:: power ::of ::noise ::at ::output}$ Channel SNR = $left ( SNR right )_C= frac{Average :: power ::of ::modulated ::signal}{Average:: power ::of ::noise ::in ::message ::bandwidth}$ Figure of Merit The ratio of output SNR and input SNR can be termed as Figure of Merit. It is denoted by F. It describes the performance of a device. $$F=frac {left ( SNR right )_O}{left ( SNR right )_I}$$ Figure of merit of a receiver is $$F=frac {left ( SNR right )_O}{left ( SNR right )_C}$$ It is so because for a receiver, the channel is the input. SNR Calculations in AM System Consider the following receiver model of AM system to analyze noise. We know that the Amplitude Modulated (AM) wave is $$sleft ( t right )=A_cleft [ 1+k_amleft ( t right ) right ] cosleft ( 2 pi f_ct right )$$ $$Rightarrow sleft ( t right )=A_c cos left ( 2 pi f_ct right )+A_ck_amleft ( t right ) cosleft ( 2 pi f_ct right )$$ Average power of AM wave is $$P_s=left ( frac{A_c}{sqrt{2}} right )^2+left ( frac{A_ck_amleft ( t right )}{sqrt{2}} right )^2=frac{{A_{c}}^{2}}{2}+frac{{A_{c}}^{2}{k_{a}}^{2}P}{2}$$ $$Rightarrow P_s=frac{{A_{c}}^{2}left ( 1+{k_{a}}^{2}P right )}{2}$$ Average power of noise in the message bandwidth is $$P_{nc}=WN_0$$ Substitute, these values in channel SNR formula $$left ( SNR right )_{C,AM}=frac{Average :: Power :: of :: AM :: Wave}{Average :: Power :: of :: noise :: in :: message :: bandwidth}$$ $$Rightarrow left ( SNR right )_{C,AM}=frac{{A_{c}}^{2}left ( 1+ {k_{a}}^{2}right )P}{2WN_0}$$ Where, P is the power of the message signal=$frac{{A_{m}}^{2}}{2}$ W is the message bandwidth Assume the band pass noise is mixed with AM wave in the channel as shown in the above figure. This combination is applied at the input of AM demodulator. Hence, the input of AM demodulator is. $$vleft ( t right )=sleft ( t right )+nleft ( t right )$$ $Rightarrow vleft ( t right )=A_cleft [ 1+k_amleft ( t right ) right ] cosleft ( 2 pi f_ct right )+$ $left [ n_1left ( t right ) cosleft ( 2 pi f_ct right ) – n_Qleft ( t right ) sin left ( 2 pi f_ct right )right ]$ $Rightarrow vleft ( t right )=left [ A_c+A_ck_amleft ( t right )+n_1left ( t right ) right ] cosleft ( 2 pi f_ct right )-n_Qleft ( t right ) sinleft ( 2 pi f_ct right )$ Where $n_I left ( t right )$ and $n_Q left ( t right )$ are in phase and quadrature phase components of noise. The output of AM demodulator is nothing but the envelope of the above signal. $$dleft ( t right )=sqrt{left [ A_c+A_cK_amleft ( t right )+n_Ileft ( t right ) right ]^2+left ( n_Qleft ( t right ) right )^2}$$ $$Rightarrow dleft ( t right )approx A_c+A_ck_amleft ( t right )+n_1left ( t right )$$ Average power of the demodulated signal is $$P_m=left ( frac{A_ck_amleft ( t right )}{sqrt{2}} right )^2=frac{{A_{c}}^{2}{k_{a}}^{2}P}{2}$$ Average power of noise at the output is $$P_no=WN_0$$ Substitute, these values in output SNR formula. $$left ( SNR right )_{O,AM}= frac {Average :: Power :: of :: demodulated :: signal }{Average :: Power :: of :: noise :: at :: Output}$$ $$Rightarrow left ( SNR right )_{O,AM}=frac{{A_{c}}^{2}{k_{a}}^{2}P}{2WN_0}$$ Substitute, the values in Figure of merit of AM receiver formula. $$F=frac{left ( SNR right )_{O,AM}}{left ( SNR right )_{C,AM}}$$ $$Rightarrow F=left ( frac{{A_{c}^{2}}{k_{a}^{2}}P}{2WN_0} right )/left ( frac{{A_{c}}^{2}left ( 1+ {k_{a}}^{2}right )P}{2WN_0} right )$$ $$Rightarrow F=frac{{K_{a}}^{2}P}{1+{K_{a}}^{2}P}$$ Therefore, the Figure of merit of AM receiver is less than one. SNR Calculations in DSBSC System Consider the following receiver model of DSBSC system to analyze noise. We know that the DSBSC modulated wave is $$sleft ( t right )=A_cmleft ( t right ) cosleft ( 2 pi f_ct right )$$ Average power of DSBSC modulated wave is $$P_s=left ( frac{A_cmleft ( t right )}{sqrt{2}} right )^2=frac{{A_{c}}^{2}P}{2}$$ Average power of noise in the message bandwidth is $$P_{nc}=WN_0$$ Substitute, these values in channel SNR formula. $$left ( SNR right )_{C,DSBSC}=frac{Average :: Power :: of :: DSBSC :: modulated :: wave}{Average :: Power :: of :: noise :: in :: message :: bandwidth}$$ $$Rightarrow left ( SNR right )_{C,DSBSC}=frac{{A_{c}}^{2}P}{2WN_0}$$ Assume the band pass noise is mixed with DSBSC modulated wave in the channel as shown in the above figure. This combination is applied as one of the input to the product modulator. Hence, the input of this product modulator is $$v_1left ( t right )=sleft ( t right )+nleft ( t right )$$ $$Rightarrow v_1left ( t right )=A_cmleft ( t right ) cos left ( 2 pi f_ct right )+left [ n_Ileft ( t right ) cosleft ( 2 pi f_ct right ) – n_Qleft ( t right ) sin left ( 2 pi f_ct right )right ]$$ $$Rightarrow v_1left ( t right )=left [ A_cm left ( t right ) +n_Ileft ( t right ) right ] cosleft ( 2 pi f_ct right )-n_Qleft ( t right ) sinleft ( 2 pi f_ct right )$$ Local oscillator generates the carrier signal $cleft ( t right )= cosleft ( 2 pi f_ct right )$. This signal is applied as another input to the product modulator. Therefore, the product modulator produces an output, which is the product of $v_1left ( t right )$ and $cleft ( t right )$. $$v_2left ( t right )= v_1left ( t right )cleft ( t right )$$ Substitute, $v_1left (