Basics of digital communications

Communication theory aims to explore and develop methods that suppress (as far as possible) the effect of noise and to simultaneously transmit as many discrete signals as possible through a channel. Spectral analysis is a tool that connects time-domain signals to the frequency domain, allowing insight into the characteristics of broadband and narrowband signals in a communication bandwidth.

Spectral analysis

Frequency has a ubiquitous role in the process of communication. It is used as a carrier and bandwidths are specified in terms of it. It is therefore important to have tools with which you can easily determine the frequency content in a signal. This can be achieved using the Fourier transform (FT) and its discrete counterpart, the DFT. The FT of a signal w(t) w(t) is defined in Equation 1 as:

W(f) =∫-∞∞w(t)exp(-j2πft)dt W(f) =∫-∞∞w(t)exp(-j2πft)dt

Its inverse is calculated with Equation 2:

w(t) =∫-∞∞W(f)exp(j2πft)df w(t) =∫-∞∞W(f)exp(j2πft)df

The above equations show that w(t) w(t) is a weighted sum of sinusoids in the interval -∞ to +∞ -∞ to +∞. The weights are complex numbers W(f) W(f). If at any particular frequency the magnitude spectrum is strictly positive, then that frequency is said to be present in w(t) w(t). The set of all frequencies present in a signal is its frequency content. If this content consists of frequencies in a certain range, then w(t) w(t) is said to be bandlimited with a certain bandwidth.

Digital modulation

It is the process of converting digital symbols into waveforms that are compatible with the characteristics of the transmission medium. In the case of baseband modulation, these waveforms take the shape of pulses designed to reduce intersymbol interference (ISI). In the case of bandpass modulation, these shaped pulses modulate a sinusoidal carrier wave that is converted to an electromagnetic (EM) field for propagation over distances. In free space, antennas radiate and receive EM signals. Antennas operate effectively only when their dimensions are of the order of magnitude of quarter wavelength ( Λ4 ) (Λ4) of the transmitted signal. If the signal frequency is very high, antenna dimension becomes practical; however, high frequencies get attenuated by the atmosphere and therefore cannot travel great distances.

Basic modulation techniques

Any message can be converted into binary digits called bits. For transmission, these bits are grouped together and encoded into sequences whose elements are the symbols of an alphabet set. To utilize bandwidth more efficiently, these alphabets are encoded in waveforms called pulses, which are then combined to form a baseband signal. For example, bitstream 01001001010010111010101 can be paired as 01 00 10 01 01 11 and so on. Then the pairs can be encoded as -1, -3, 1, -1, +3 and so on to produce the symbol sequence. There are many ways to map from bits to symbols. Bitstreams can be mapped to eight-level, 16-level, 256-level, etc. After the original message is grouped into alphabets, it must be turned into analog waveforms by choosing a pulse shape p(t) p(t) and then transmitting –p(t - kT), -3p(t – kT), p(t – kT), -p(t – kT), 3p(t –kT) –p(t - kT), -3p(t – kT), p(t – kT), -p(t – kT), 3p(t –kT). In general, this four-level signal takes the form shown in Equation 3:

d(t) = ∑k Ik p(t - kT); Ik Є{±1, ±3}; k = 1, 2, ... d(t)=∑kIkp(t - kT); IkЄ{±1, ±3}; k = 1, 2, ...

Where:

d(t)d(t) = an analog waveform consisting of pulses at symbol time kTkT and the amplitude of the pulse is proportional to the associated symbol value.

Ideally, the pulse should be chosen so that the value of message at kk does not interfere with the message at any other time (no ISI) and makes efficient use of bandwidth.

ISI can manifest itself in two ways: when the pulse shape p(t)p(t) is wider in time than a single symbol time interval TT, and when the pulses experience channel distortions and multipath fading effects.

Consider a two-level system in which s0 (t)s0(t) and s1(t)s1(t) are finite energy signals representing logical 0 and 1, respectively. These signals can be of any shape but must have finite energy in the signaling interval. Then you can construct a framework of representative basic modulation schemes. Some fundamental digital modulation schemes are below.

Amplitude shift keying

In amplitude shift keying (ASK), the information is conveyed by varying the amplitude of a carrier wave in accordance with the symbol stream. ASK can be expressed as Equation 4:

sk (t) = Ik p(t)cos( w0 t+Ф); Ik Є{0,1}; 0≤t≤T ; sk(t)=Ikp(t)cos(w0t+Ф); IkЄ{0,1}; 0≤t≤T ;

In Equation 4, the phase term is an arbitrary constant. Binary ASK signaling, also called on-off keying, was one of the earliest forms of digital modulation used in radio telegraphy. ASK has a high peak-to-average ratio and is no longer widely used; however, TI’s low-power wireless radio-frequency integrated circuits support this modulation scheme for various data rates in sensor applications. Figures 1 - 4 shows waveforms for Amplitude Shift Keying.

Figure 1. Amplitude Shift Keying source symbols.

Figure 2. Amplitude Shift Keying modulated symbols.

Figure 3. Amplitude Shift Keying constellation plot.

Figure 4. Amplitude Shift Keying spectrum.

Frequency-shift keying

The general analytical expression for frequency shift keying (FSK) is given in Equation 5:

sk (t) = A cos( w0 t+2π Ik Δft+Ф); k = 1,...M; 0≤t≤T sk(t)=Acos(w0t+2πIkΔft+Ф); k = 1,...M; 0≤t≤T

Where:

The frequency Δf = the amount of shift in the carrier frequency corresponding to the alphabet I_k ε{±1, ±3. . .,±M}

Phase term is an arbitrary constant.

The FSK waveform sketch in the following figures show typical frequency changes at the symbol interval. The change from one frequency to another can be rather abrupt; this gives rise to spikes in the spectrum of FSK. The minimum required bandwidth for orthogonal FSK signals for coherent detection is 1/2T, whereas for noncoherent detection the bandwidth is 1/T. FSK does not have constellation plots because of constant rotation of the signal vector in the IQ plane. Figures 5 - 7 are typical waveforms for FSK.

Figure 5. Frequency Shift Keying symbol source.

Figure 6. Frequency Shift Keying modulated symbols.

Figure 7. Frequency Shift Keying spectrum.

Phase-shift keying

Developed the during early days of the space program, phase-shift keying (PSK) is widely used in commercial satellite links. The general expression for PSK is shown in Equation 6:

sk (t) = p(t)cos( w0 t+ Фk ); k = 1,...,M 0≤t≤T sk(t)=p(t)cos(w0t+Фk); k = 1,...,M 0≤t≤T

Where the phase term ФiФi will have M discrete values typically given by Equation 7:

Фk(t) = 2πkM k = 1,...M Фk(t) = 2πkM k = 1,...M

In the basic case of binary PSK, the modulating data signal shifts the phase of the waveforms to one of two states: either zero or ππ. The waveform sketch in the figure shows abrupt phase changes in the signaling interval. Figures 8 through 11 are typical waveforms for a PSK modulation system.

Figure 8. Phase Shift Keying symbol source.

Figure 9. Phase Shift Keying modulated symbols.

Figure 10. Phase Shift Keying constellation plot.

Figure 11. Phase Shift Keying spectrum.

Comparison metric for digital communication and BER curves

One of the most important metrics of performance in digital communication is the plot of bit error probability PbPb versus energy per bit over noise power spectral density ( Eb N0Eb N0 ). This dimensionless ratio is a standard quality measure for digital communication system performance. The smaller the Eb N0Eb N0 required, the more efficient the detection process for a given probability of error. In digital communication systems, one discrete symbol is transmitted that may be one bit or more in a fixed signaling interval. In analog, where the information source is continuous, this discrete structure does not exist. In digital systems, a figure of merit should allow us to compare one system with another at a bit or symbol level; hence Eb N0Eb N0 renders itself naturally for that purpose. A relationship between the SNR, data rate and bandwidth is expressed in Equation 8:

Eb N0 = SN ( BR ) Eb N0=SN(BR)

Figures 12 through 14 show typical PbPb vs. Eb N0Eb N0 curves for orthogonal and multiphase signaling.

Figure 12. The curve to the left in the above chart represents M = 32 while the curve to the right represents M = 2.

Figure 13. The curve to the left in the above chart represents M = 2 and the curve to the right represents M = 32.

Figure 14. BER comparisons of ASK, 2FSK and 2PSK

Digital modulation methods can be classified in two ways, with opposite behavioral characteristics. The first class is orthogonal signaling; its error performance follows the curves in the first figure. The second class constitutes nonorthogonal signaling is shown in the second figure. Error performance improvement or degradation depends on signaling category.

Channel capacity

Claude Shannon's fundamental theorem states that it is possible (in principle, using some coding scheme) to transmit information with an arbitrarily small probability of error, provided that the data rate RR is less than or equal to the channel capacity CC . Shannon’s work showed that SNR and bandwidth set a limit on transmission rate but not on probability of error. The channel capacity of a white bandlimited Gaussian noise channel is expressed in Equation 9:

C = Blog2 ( 1+ Eb N0 BR ) bits/sec/HzC = Blog2(1+Eb N0BR)bits/sec/Hz

Where:

BB = the channel bandwidth

Eb N0Eb N0 = the energy/bit with an example SNR of 30 dB

Using Equations 8 and 9, the capacity of a circuit with 2.4-kHz bandwidth is approximately 24 kbps, whereas at 10-dB SNR the capacity drops to about 8.3 kbps. Thus, Shannon's theorem allows designers to apply trade-offs in bandwidth, signal power and various modulation methods to establish a communication link with a desired probability of error.

Similarly, the required Eb N0Eb N0 for a modem operating at a channel capacity of 28.8 kbps in an AWGN bandwidth of 3.4 kHz will be approximately 16.3 dB.

Bandwidth and power constraints

The design of a digital communication system begins with the channel description, received power, available bandwidth, noise statistics, and definition of system requirements such as data rate and error performance. Two primary communication criteria are the received power and available bandwidth. In bandwidth-limited systems, spectrally efficient schemes can save bandwidth at the expense of power. In power-limited systems, power-efficient schemes can be used at expense of bandwidth.

For any digital communication system, the relationship between received power to noise-power spectral density Pr N0Pr N0 and received Eb N0Eb N0 is given by Equation 10:

Pr N0 = Eb N0R Pr N0=Eb N0R

Where N0= NBN0=NB . This relationship is frequently used in designing and evaluating digital communication systems.

Bandwidth-limited systems

Bandwidth efficiency increases as B Tb BTb product decreases. Therefore, signals with small B Tb BTb products are employed in bandwidth-limited systems. In uncoded systems, the objective is to maximize the information rate within the allowable bandwidth at the expense of Eb N0Eb N0 while maintaining a required PbPb . MPSK and MQAM are examples of bandwidth-efficient modulation schemes with bandwidth efficiency (Equation 11):

RB = log2 ( M ) bits/sec/HzRB = log2(M)bits/sec/Hz

Suppose you have to choose between MFSK and MPSK for the following parameters: bandwidth = 4 kHz, data rate = 10 kbps and Pr N0Pr N0 = 60 dB-Hz. First you find that the received Eb N0Eb N0 = 55 – 10*log10(10000) = 15 dB. Since the required data rate exceeds the bandwidth required, the best choice is MPSK. Next, decide on the value of M that will give a symbol rate closest to the bandwidth of 4 kHz. From M = 8, you know that the symbol rate is 3.2 kHz. Next, you'll see that for PbPb of less than 10e-5, the required bandwidth is around 13 (from BER curves) which is less than the received, so the best choice is 8PSK.

Power-limited systems

For this type of system, where power is limited but bandwidth is abundant, the following trade-offs are possible:

1. Improved Pb Pb at the expense of bandwidth for fixed Eb N0Eb N0 .

2. Reduction in Eb N0Eb N0 at the expense of bandwidth for fixed PbPb .

MFSK is an orthogonal signaling technique used in power-limited systems. It has a bandwidth efficiency of noncoherent MFSK given by Equation 12:

RB = log2(M) M bits/sec/HzRB = log2(M) M bits/sec/Hz

Suppose you have an available bandwidth of 45 kHz and Pr N0Pr N0 = 50 dB-Hz. Again the goal is to choose a modulation scheme to meet the same BER performance. The received Eb N0Eb N0 = 50 – 10log(10000) = 10 dB. Since you have plenty of bandwidth compared to the data rate, the best modulation scheme would be MFSK. In an effort to conserve power, you should look for the largest M such that the minimum bandwidth for MFSK doesn’t exceed 45 kHz. For M = 16, the required Eb N0Eb N0 to keep PbPb less than 10e-5 is around 8 dB (from BER curves), which is below 8.2 dB.

References

1. Sklar, Bernard. Digital Communications: Fundamentals and Applications. Prentice Hall, 2001. http://books.google.com/books/about/Digital_communications.html?id=Bh4fAQAAIAAJ

2. Molisch, Andreas F. Wireless Communications. John Wiley and Sons, 2010. http://books.google.com/books/about/Wireless_Communications.html?id=vASyH5-jfMYC

3. Johnson, C. Richard, and Sethares, William A. Telecommunication Breakdown: Concepts of Communication Transmitted Via Software-Defined Radio. Prentice Hall, 2004.

4. Gu, Qizheng. RF System Design of Transceivers for Wireless Communications. Springer, 2005. http://books.google.com/books?id=fuUwM1Hiu24C

Content actions

Add module to:

My Favorites Login Required (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own Login Required (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Footer

This work is licensed by farrukh inam and Thomas Almholt under a Creative Commons Attribution License (CC-BY 3.0), and is an Open Educational Resource.

Last edited by Gene Frantz on Mar 18, 2013 3:31 pm -0500.