AWGN channel | why awgn channel is used in communication | power spectral density of awgn channel

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SHORT QUESTIONS WITH ANSWERS
Q.1.     Explain the concept of AWGN channel.
Ans.    A channel is assumed to have following two characteristics:
(i)         the channel is linear with a bandwidth that is wide enough to accommodate the transmission of signal s_i(t) with negligible or no distortion.
(ii)        the channel noise , w(t), is the sample function of a zero-mean white Gaussain noise process.
At this stage, it may be noted that the reasons for second assumption are that it makes receiver calculations very easy and also, it is a reasonable description of the type of noise present in several practical communication systems. Such a channel is popularly known as an additive white Gaussian noise (AWGN) channel.
Hence, in view of above discussion, the received signal x(t) is expressed as
Q.2.     Explain the concept of Optimum Receiver.
Ans.    the receiver observes the received signal x(t) for a duration of T seconds and makes a best estimate of the transmitted signal s_i(t) or equivalently the estimate of symbols m_i. But, due to the presence of channel noise, this decision-making process is statistical in nature. As a result of this, the receiver is likely to make occasional errors.
Therefore, the requirement is to design the receiver so as to minimize the average probability of symbol error.
This average probability of symbol error may be defined as
where,                         m_i = transmitted symbol,
= estimate produced by the receiver, and
= the conditional error probability given that the ith symbol was sent.
Q.3.     What is Geometric Representation of Signals? Explain.
Ans.    In geometric representation of signals, we represent any set of M energy signals {s_i(t)} as linear combinations of N orthonormal basis functions, where N ≤ M.
This means that given a set of real-valued signals s₁(t), s₂(t), …, s_M(t), each of duration T seconds, we may write s_i(t) as under:
equation
where, the coefficients of the expansion can be defined as
equation
Now, the real-valued basis functions ₁(t) ₂(t), …, _N(t) are orthonormal. Here, the word ‘orthonormal’ implies that
equation
Where _ij the Kronecker delta.
Q.4.     Explain Schwarz Inequality
Ans.    Let us consider any pair of energy signals si(t) and s2(t). The Schwarz inequality states that
EQUATION
The equality holds if and only if s₂(t) = cs₁(t), where c is any constant.
The proof of the Schwarz inequality applies to real-valued signals. It may be readily extended to complex-valued signals, in which case above expression can be reformulated as under:
EQUATION
where the equality holds if and only if s₂(t) = cs₁(t), where c is a constant.
QUESTIONS

1. Draw the block diagram of a most basic form of digital communication system and write the expression for probable that symbol m_i is emitted by an information source.
2. Explain the concept of AWGN channel.
3. What is the concept of an optimum receiver? Explain.
4. Explain the geometric representation of signals.
5. What is Gram-Schmidt orthogonalization procedure? Explain.

DIGITAL MODULATION TECHNIQUES
Inside this Chapter

• Introduction
• Digital Modulation Formats
• Types of Digital Modulation Techniques
• Coherent Binary Modulation Techniques
• Coherent Binary Amplitude Shift Keying or On-Off Keying
• Binary Phase Shift Keying
• Coherent Binary Frequency Shift Keying
• Non-Coherent Binary Modulation Techniques
• Non-Coherent Binary Amplitude Shift Keying
• Non-Coherent Detection of FSK
• Differential Phase Shift Keying
• Duadrature Phase Shift Keying (QPSK)
• Generation of QPSK
• Minimum Shift Keying
• Comparison of Digital Modulation Techniques

8.1       INTRODUCTION
As discussed earlier, Modulation is defined as the process by which some characteristics of a carrier is varied in accordance with a modulating signal. In digital communications, the modulating signal consists of binary data or an M-ary encoded version of it. This data is used to modulate a carrier wave (usually sinusoidal) with fixed frequency. In fact, the input data may represent the digital computer outputs or PCM waves generated by digitizing voice or video signals. The channel may be a telephone channel, microwave radio link, satellite channel or an optical fiber. In digital communication, the modulation process involves switching or keying the amplitude, frequency or phase of the carrier in accordance with the input data.
Thus, there are three basic modulation techniques for the transmission of digital data. They are known as amplitude-shift keying (ASK), frequency shift keying (FSK) and phase-shift keying (PSK) which can be viewed as special cases of amplitude modulation frequency modulation and phase modulation respectively.

The present chapter is devoted to detailed discussion of digital modulation techniques: their noise performance, spectral properties, their merits and limitations, applications and other related aspects.
8.2       DIGITAL MODULATION FORMATS
When we have to transmit a digital signal over a long distance, we need continuous-wave (CW) modulation. For this purpose, the transmission medium can be in form of radio, cable or other type of channel. Also, a carrier signal having some frequency f_c is used for modulation. Then the modulating digital signal modulates some prameter like frequency, phase or amplitude of the carrier. Due to this process, there is some deviation in carrier frequency f_c. This deviation is known as the bandwidth of the channel. This means that the channel has to transmit some range or band of frequencies. Such type of transmission is known as bandpass transmission and the communication channel is known as bandpass channel.
            Here, the word bandpass is used since the range of frequencies does not start from zero Hz to fm Hz. In fact, the range of frequencies from zero Hz to fm Hz is known as low-pass signal and such channel is known as low-pass channel.
            Now, when it is required to transmit digital signals on a bandpass channel, the amplitude, frequency or phase of the sinusoidal carrier is varied in accordance with the incoming digital data. Since the digital data is in discrete steps, the modulation of the bandpass sinusoidal carrier is also done in discrete steps. Due to this reason, this type of modulation (i.e., Digital modulation) is also known as switching or signaling. Now, if an amplitude of the carrier is switched depending on the input digital signal, then it is called Amplitude shift keying (ASK).
This process is quite similar to analog amplitude modulation. If the frequency of the sinusoidal carrier is switched depending upon the input digital signal, then it is known as the frequency shift keying (FSK). This is very much similar to the analog frequency modulation. If the phase of the carrier is switched depending upon the input digital signal, then it is called phase shift keying (PSK). This is similar to phase modulation. Since the phase and frequency modulation has constant amplitude envelope, therefore FSK and PSK also has a constant amplitude envelope. Because of constant amplitude of FSK and PSK, the effect of non-linearities, noise interference is minimum on signal detection. However, these effects are more pronounced on ASK. Therefore, FSK and PSK are preferred over ASK.
Figure 8.1 shows the waveforms for amplitude-shift keying, phase-shift keying and frequency shift keying. In these waveforms, a single feature of the carrier (i.e., amplitude, phase or frequency) undergoes modulation.
DIAGRAM
FIGURE 8.1 The three basic forms of signaling binary information,
(a)        Amplitude-shift keying,          (b)        Phase-shift keying,
(c)        Frequency shift keying with continuous phase
In digital modulations, instead of transmitting one bit at a time, we transmit two or more bits simultaneously. This is known as M-ary transmission. This type of transmission results in reduced channel bandwidth. However, sometimes, we use two quadrature carriers for modulation. This process is known as Quadrature modulation.
            Thus, we see that there are a number of modulation schemes available to the designer of a digital communication system required for data transmission over a bandpass channel.
Every scheme offers system trade-offs of its own. However, the final choice made by the designer is determined by the way in which the available primary communication resources such as transmitted power and channel bandwidth are best exploited. In particular, the choice is made in favour of a scheme which possesses as many of the following design characteristics as possible:
(i)         Maximum data rate,
(ii)        Minimum probability of symbol error,
(iii)       Minimum transmitted power,
(iv)       Maximum channel bandwidth,
(v)        Maximum resistance to interfering signals,
(vi)       Minimum circuit complexity.
8.3       TYPES OF DIGITAL MODULATION TECHNIQUES
(U.P. Tech., Sem. Examination, 2003-2004)
Basically, digital modulation techniques may be classified into coherent or non-coherent techniques, depending on whether the receiver is equipped with a phase-recovery circuit or not. The phase-recovery circuit ensures that the oscillator supplying the locally generated carrier wave receiver is synchronized* to the oscillator supplying the carrier wave used to originally modulate the incoming data stream in the transmitter.
* In both frequency and phase.

 (i)        Coherent Digital Modulation Techniques
            Coherent digital modulation techniques are those techniques which employ coherent detection. In coherent detection, the local carrier generated at the receiver is phase locked with the carrier at the transmitter. Thus, the detection is done by correlating received noisy signal and locally generated carrier. The coherent detection is a synchronous detection.
(ii)        Non-coherent Digital Modulation Techniques
            Non-coherent digital modulation techniques are those techniques in which the detection process does not need receiver carrier to be phase locked with transmitter carrier.
The advantage of such type of system is that the system becomes simple. But the drawback of such a system is that the error probability increases.
In fact, the different digital modulation techniques are used for various specific application areas.
8.4       COHERENT BINARY MODULATION TECHNIQUES
As mentioned earlier, the binary (i.e., digital) modulation has three basic forms amplitude-shift keying (ASK), phase-shift keying (PSK) and frequency-shift keying (FSK). In this section, let us discuss different coherent binary modulation techniques.
8.5       COHERENT BINARY AMPLITUDE SHIFT KEYING OR ON-OFF KEYING
(i)         Definition
            Amplitude shift keying (ASK) or ON-OFF keying (00K) is the simplest digital modulation technique. In this method, there is only one unit energy carrier and it is switched on or off depending upon the input binary sequence.
Expression and Waveforms
The ASK waveform may be represented as,
s(t) =  cos (2f_ct) (To transmit ‘1’)                    …(8.1)
diagram
FIGURE 8.2 Amplitude-shift keying waveforms, (a) Unmodulated carrier,
(b) NRZ Unipolar bit sequence, (c) ASK waveform.
To transmit. symbol ‘0’, the signal s(t) = 0 i.e., no signal is transmitted. Signal s(t) contains some complete cycles of carrier frequency ‘f’_c.
Hence, the ASK waveform looks like an ON-OFF of the signal. Therefore, it is also known as the ON-OFF keying (00K). Figure 8.2 shows the ASK waveform.
8.5.1. Signal Space Diagram of ASK
            The ASK waveform of equation (8.1) for symbol ‘1’ can be represented as,
s(t) = .  cos (2f_ct) = (t)
This means that there is only one carrier function (t). The signal space diagram will have two points on (t). One will be at zero and other will be . Figure 8.3 shows this aspect.
diagram
FIGURE 8.3 Signal space diagram of ASK.
Thus, the distance between the two signal points is,
d =  =                                          …(8.3)