# Wireless Networking Fundamentals

06

## Analog and digital signals. Modulation

### Analog and digital signals

Signals distributed in any environment, are continuous in values and time. Signals that are continuous in values of voltage and time are called analog (Figure 1(a)). The main reason for the mass refusal of analog signals in favor of digital in wireless communication systems is the development of digital technology that operates with numerical data, rather than analog signals. Also, advantages of digital signals are the possibility of recovery in intermediate transmission nodes and the coding tool used to recover lost data and solve security issues.

If the signal samples are fixed at the selected time instants, then such a signal is called discrete, and the operation for obtaining it from the analog is called sampling (Figure 1(b)). To make analog-signal sampling the Kotelnikov theorem should be used, according to which the sampling frequency should be at least twice higher than frequency of the signal spectrum. In case if the Kotelnikov theorem is not fulfilled, then some transmitted in the signal data will be lost. So when transmitting a signal in the 5.12 - 5.32 GHz band, the sampling rate should be not lower than 10.64 GHz; when transmitting a signal in the 2.432 - 2.452 GHz band, the sampling frequency should be not lower than 4.904 GHz.

If signal voltage values are reduced to several final levels, then the received signal is called quantized, and the operation for obtaining it from the analog is called quantization (Figure 1(c)).

The result of two operations, sampling and quantization, over an analog signal is called a digital (Figure 1(d)).

Figure 1 Time dependencies of a harmonic signal in various representations: a - analog, b - discrete, c - quantized, d - digital

Using digital signals, it is possible to represent the original analog signal in the form of a finite numerical sequence, which make storaging and data processing easier, and also make possible to use coding schemes.

Let's look at the signal presented in Figure 1(d), to explain the coding technique. To quantize the signal, 8 levels are used, three bits for the numerical designation of each level in the binary system. At zero time, the signal value is at the level "100", the same level - "100" is in next time reckoning, then in following time reckoning the signal value increases and becomes "101", the signal increases again in next time reckoning at the level "110", etc. The final signal record in digital form is shown in Figure 2, the original analog signal is represented as a sequence "100100101110111111110110100100".

Figure 2 Part of harmonic signal recorded in digital form

Keep in mind that presentation of an analog signal in digital form, leads to the lost of some signal information, the similarity of signals will be determined by the sampling rate and the number of quantization levels. Music codecs use same principle: the audio stream can be encoded by different codecs, which affect the size and the subjective playback quality. Thus, when converting an analog signal to digital, consider the amount of output data and the importance of the data lost during conversion.

An important advantage of digital signals is the possibility of their reproduction: the received analog signal can be amplified in amplitude, however, interference will be also amplified, while the digital signal can be first decoded and then generated again.

### Coding

Coding is a transformation of a numerical sequence into another numerical sequence, usually of a longer length. Coding is used to detect and / or correct errors that occurs during data transmission.

Let's look at the simplest kind of coding with parity check: in order to do this write the numerical sequence obtained in the previous paragraph, element-wise to the column:

Figure 3 Encoding of numerical sequence with a parity check code

Add an additional bit to each symbol, zero if the sum of bits is even, and one if the sum of bits is odd. Thus, the sum of bits of the new four-digit number is even.

During transmission, the sequence modulates the carrier signal. On the receiving side, the received signal is decoded and divided into 4-bit sequences, each of which encodes one symbol. Each of four-bit numbers is checked for parity, if the sum of bits is odd, then an error is recorded in the symbol transmission. A parity check code can only detect one error, if two of four bits are incorrectly received, the error will not be detected. This encoding method is simple enough, more complex code constructions allow not only to detect, but also to correct one or more errors.

### Modulation

A signal containing information occupies a certain band, is located in the low frequency range and is called modulating. A high-frequency signal, one or more parameters of which varies according to a modulating signal is called modulated. The process of transferring the spectrum of the information signal into the high-frequency range, is called modulation. The using of such conversion has two advantages:

• decreases dimensions of receiving and transmitting antennas, as was shown in section 3. Antennas;
• the transfer of the signal spectrum to a certain frequency makes possible to realize some multiple access schemes, such as FDMA, OFDMA, showed in 5. Multiple Access Methods.

#### Modulation scheme

The modulation process is the amplitude, frequency or phase changing of the carrier frequency in accordance with the incoming data.

Let's look at Figure 4 to explain the modulation process. The carrier generator forms a reference signal, which is a predetermined frequency harmonic signal entering the modulator input. The transmitted message source forms a bit stream, which must be transmitted to the receiving side. In accordance with the generated bit stream, one or more reference signal parameters are changed in the modulator block, the received signal is transmitted to the next stages and is radiated to the receiver.

Figure 4 Modulation scheme

#### Constellation diagram IQ

Constellation diagram is a tool for the signal modulation scheme evaluation, represents a radio signal in the two-dimensional point diagram form on the complex plane. Any signal can be represented as a sum of in-phase (I) and quadrature (Q) components and is shown as a point on the complex plane. The set of signals generated by transmitter and get distorted during the radiation, is detected on the receiving side, forms a set of points on the complex plane, which is called a constellation diagram.

Figure 5 The constellation diagram coordinate system

For example, let's look at system with binary phase shift keying (BPSK), shown below more detaily. For transmission the "1" symbol a signal with zero initial phase is used, for the "0" symbol - a signal shifted to π, relative to the first, so only two signals are allowed in the system, when receiving a signal with a phase value different from allowed, the signal will be assigned to one of preset. Thus, the constellation diagram for this example is:

Figure 6 Constellation diagram for BPSK (FM-2)

Let's look at an example of data flow transmission "10110100" using different types of modulation. There is difference between modulation of analog and digital signals - the modulation of digital signals is called manipulation. Further examples will be taken for digital signals, so the shift keying term will be used.

In amplitude shift keying, each of transmitted symbols is associated with a different amplitudes high-frequency signal. At Figure 5 (b) to the transmitted symbol "1" is assigned to a signal with an amplitude 1, and to the symbol "0" - with an amplitude of 0.5. In Figure 5(c) more signal amplitudes are used, it makes possible to transmit two information symbols in message signal unit: signal with an amplitude 1 is assigned to symbols "11", 0.75 - with an amplitude "10", 0.5 - with an amplitude "01", 0.25 - with an amplitude "00".  Thus, the selected modulation scheme affects the number of information symbols transmitted within single symbolic package. The number of signal amplitude levels depends on the transmitting and receiving side hardware capabilities, as well as on the communication channel characteristics and directly affects the data rate. So, with the same interval of a message signal unit, using of a four-level amplitude shift keying (ASK-4) doubles the data transfer rate in compare to the binary amplitude shift keying (ASK-2). This example is shown in Figure 7: the original sequence is transmitted in four packages with using of ASK-4 (Figure 7 (c)), and in eight when using ASK-2 (Figure 7 (b)). Please note, other amplitude shift keying types ex. ASK-8, ASK-16 could also be used.

Figure 7 Amplitude shift keying oscillograms: a - source data sequence, b - ASK-2, c - ASK-4

#### Frequency shift keying (FSK)

In frequency shift keying, each of transmitted symbols is associated with a different frequency of high-frequency signal. So, two frequencies signals are used for binary FSK-2 frequency shift keying, and four frequencies signals for FSK-4. Similarly to amplitude shift keying, the number of signals used affects the transmission speed and depends on the equipment hardware capabilities same as on the communication channel quality. In Figure 8 are shown FSK-2 modulated signal oscillograms:

Figure 8 Frequency shift keying oscillogramsa - source data sequence, b - signal corresponding to a "1" symbol, c - signal corresponding to a "0" symbol, d - FSK-2 signal

#### Phase shift keying (PSK)

In phase shift keying, transmitted symbols are associated with high frequency signals with different initial phase values. An example of using PSK-2 for the data sequence shift keying is shown in Figure 9: the symbol "1" corresponds to a signal with a zero initial phase, the symbol "0" - a signal with a π phase value. Characteristics of PSK-4 and PSK-8 shift keying are similar to amplitude and frequency shift keying.

Figure 9 Phase shift keying oscillogramsa - source data sequence, b - signal corresponding to a "1" symbol, c - signal corresponding to a "0" symbol, d - PSK-2 signal

In previous shift keying types, only one parameter of the high-frequency signal was changed: amplitude, frequency or phase. Quadrature amplitude manipulation uses a combination of different amplitude levels and phase shifts that correspond to transmitted bits of information. So, using of QAM-16 allows 4 amplitude values and 4 phase shifts, the combination "each with each" gives 16 possible signal versions - points on the signal constellation:

Figure 10 Signal constellation QAM-16

Let's look at the example of QAM-16 using. In this case modulated signal oscillograms are following:

Figure 11 QAM-16 oscillograms

#### Orthogonal frequency division multiplexing (OFDM)

One of the propagation problems, shown in section 2. Radio signal propagation fundamentals, is a multipath effect and as a resault intersymbol interference (ISI). OFDM helps to reduce ISI effect by dividing the frequency range on many subcarriers, each subcarrier uses lower modulation with a large guard interval while maintaining the overall transmission rate. In addition, OFDM allows to increase the system stability to frequency selective fading, because it does not effect the whole signal spectrum, but only few subcarrier. Disadvantage of this method is the sensitivity to the Doppler effect.

Let's look at the example of using 8 OFDM sub-carriers modulated by QAM-16 as it is shown in Figure 12. Since QAM-16 is the type of shift keying of each subcarrier, one symbolic package corresponds to the transmission of 4 bits of information and the information bitstream must be divided into 4-bit blocks. On the next step, to each of the four-bit blocks is assigned the phase and amplitude of the high-frequency signal - a symbolic package. After that, the symbol stream (not to be confused with the source data bit stream) is distributed into 8 parallel channels in accordance with number of subcarriers.

Figure 12 The bitstream distribution over eight channels with QAM-16 shift keying

After that each stream is transferred to a given frequency and transmitted to the air:

Figure 13 The streams distribution over the subcarriers in the radio channel

After reverse conversions are performed on the receiver side, the phase and amplitude of the each subcarriers signal are estimated. After the signal parameters estimation, it is mapped to the given signal constellation - the orange dot in Figure 14(a). Since the signal parameters have changed during propagation, it does not accurately correspond to the signal constellation allowed position, however, in this case, it supposed that the sequence "0001" is transmitted, since it is the closest. Thus, it is possible to determine areas in the constellation around each signal dot, which will bring received signal to allowed value. For example, square areas can be used, as in Figure 14(b).

Figure 14 Corresponding of the received signal

to the constellation on the receiving side, QAM-16 (a and b) and QAM-4 (c)

However, in case of strong signal distortion when another sequence was transmitted instead, for example "0011", then an error occurs. In wireless communication, the connection quality is permanently monitored by transmission of service messages, in case of the described situation, the system can reduce the modulation to QAM-4, for example. It will reduce the number of errors in transmission but the speed will also be reduced. The signal constellation after the modulation change is shown in Figure 14 (c). Reducing of the modulation leads to the increasing of area around each allowed dot, it directly affects the level of transmission errors.

### Bitrate

The bitrate is directly related to the coding method and the shift keying method used. Let's look at the example, one frequency channel with QAM-64 5/6 manipulation is used, i.e. one symbolic package corresponds to six bits of data. In addition, when the original sequence is encoded 1 redundant bit is added to each 5 bits of the original sequence, the duration of one symbolic package is 1 μs, then: