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Modulation Schemes

Introduction

This page will give an overview of the modulation schemes which could be used for SEDSAT-2 Communications. It builds and expands on the results from the discussion at SEDSIC'07, as noted in the SEDSAT-2 Communications Design Notes .

Modulation schemes work by taking a carrier signal eg. a cosine wave at frequency $ω c = 400 M H z$ and altering some of its parameters. If the full mathematical expression for a cosine wave is written out as a function of time, we have:

$s (t) = A c o s ((ω + Δω) t + θ)$

where f is the signal amplitude at time t, A is the signal scaling factor, $ω$ is the signal frequency in radians ( $ω = 2π f$ ), $Δω$ is the change in frequency and $θ$ is the phase of the signal.

An example: if we set A=1, $ω = 2 * p i * 433 M H z$ , $Δω = 0$ and $θ = 0$ and we plot s(t) as a function of t, we have a pure cosine wave at 433MHz. This is just an unmodulated 433MHz carrier signal with no information/data added to the signal. eg: .

Modulation schemes work by varying one of the variables in the $s (t)$ equation above. We either vary the variables continuously (in which case the modulation is analog), or we move the variables in steps (in which case the modulation is digital). For analog modulation schemes, modulating A gives us Amplitude Modulation (AM). If we modulate $Δω$ continuously, we have Frequency Modulation (FM). And finally, if we modulate $θ$ , we have phase modulation.

For SEDSAT we are transmitting data which is digital, so we may want to look into a digital modulation scheme. For these, we usually use so-called binary modulation, where we just alternate the $A$ , $Δω$ and $θ$ variables between two values. This will be discussed below.

Comparing modulation schemes

Modulation schemes are usually compared using two criteria: spectral efficiency and power efficiency.

Two additional characteristics of modulation schemes, which are both related to the $P e$ , are: spectral efficiency and power efficiency.

Spectral efficiency refers to how well the modulation can can make use of some amount of bandwidth. Bandwidth is related to the data transfer rate you can achieve, and some modulation schemes can achieve a higher data rate than others for the same bandwidth.

Power efficiency refers to how easily a signal can be detected. Some modulation schemes are relatively easy to be detected, so they have a high power efficiency. The result of this is that you only need a small SNR at the receiver to receive the signal, which means you don't need as much transmit power, as much antenna gain, etcetera, as a modulation scheme with a poorer power efficiency requires.

The power efficiency of digital modulation schemes is usually compared by looking at the probability of error, $P e$ , of the modulation scheme. This is just the likelihood that some data transmitted by the modulation scheme will be received in error at the receiver, due to noise in the communication channel. The $P e$ of a modulation scheme is expressed as a function of the energy required to transmit one bit of data, $E b$ and the noise in a 1Hz bandwidth, $N 0$ . Hence we have the famous ratio: $\frac{E_{b}}{N_{0}}$ .

This ratio is how the relative signal and noise power is written when comparing modulation schemes. However, it is not as well understood as the Signal to Noise Ratio of a signal, which gives the same information. The ratio can be converted into a Signal to Noise ratio as follows. The expression which relates power to energy is: $P = E / T$ , so re-writing this in terms of energy we have: $E b = P / T s$ where P is the signal power and $T s$ is the period of one data input symbol (eg. a 1 or a 0). Similarly, the noise power of 25mW in a 25kHz can be transformed into the noise in a 1Hz bandwidth by dividing the noise power by the signal bandwidth - in this example, $N 0 = 1 e - 6 W / H z$ .

In summary, the ratio $\frac{E_{b}}{N_{0}}$ is related to the Signal to Noise ratio (P/N) of a signal by:

$\frac{E_{b}}{N_{0}} = \frac{P T_{0} B}{N_{0}} = \frac{P}{N}T_{0}B$

Other ways of comparing modulation schemes include comparing them based on how complex the transmitter and receivers have to be. Generally, modulation schemes which are more spectrally efficient are more complex to build.

Digital modulation

Now, an overview of the binary digital modulation schemes.

Binary Amplitude Shift Keying (BASK)

BASK involves adjusting the amplitude of the carrier to match a binary data stream that we want to transmit. Typically, the amplitudes used are A=1 for a binary 1 in the input data stream and A=0 for a binary 0 in the input data stream, an as such, this specific case (when one amplitude is zero) is often called on-off-keying (OOK). For example, if we want to transmit the bitstream 10101100 using BASK, the transmitted signal would look like the following:

The probability of error of BASK is as follows:

$P_{e} = \frac{1}{2}\left[1 - erf \left( \frac{1}{\sqrt{2}} \left( \frac{E_{b}}{N_{0}} \right)^{\frac{1}{2}} \right) \right]$

The bandwidth of a BASK signal is usually taken as a

Binary Frequency Shift Keying (BFSK)

BFSK involves adjusting the frequency of the carrier to match a binary data stream that we want to transmit. The frequency spacing used will depend on how much noise there is in the system - if there is a lot of noise, two closely spaced frequencies might merge together and make correctly receiving the information impossible. If the frequencies are space too far apart, design of the system becomes very hard. Using the previous example of transmitting the bitstream 10101100 but this time using BFSK, we have the following transmitted signal:

The probability of error of BFSK is as follows:

$P_{e} = \frac{1}{2}\left[1-erf\left(\frac{1}{\sqrt{2}}\left(\frac{E}{N_{0}}\right)^{\frac{1}{2}}\right)\right]$

Binary Phase Shift Keying (BPSK)

BPSK involves adjusting the phase of the carrier to match a binary data stream that we want to transmit. The two phases usually used are $0^\circ$ and $180^\circ$ , as these phases are the furthest phase-distance apart possible.

The probability of error of BPSK is given by: $P_{e} = \frac{1}{2} \left[ 1 - erf \left( \left( \frac{E_{b}}{N_{0}} \right)^{\frac{1}{2}} \right) \right]$

Results & Summary

The $P e$ equations provide a method of comparing the different modulation schemes. If the $P e$ of each modulation scheme is plotted versus the Eb/No ratio, the we can see which modulation scheme can provide the lowest error rate for a given signal quality. This has been done:

Note that the lines for BASK and BFSK overlap so BASK is not visible.

From this information, it appears that BPSK would be a better modulation scheme for SEDSAT-2 than BASK or BFSK, as for a given error rate (ie. data rate), it requires a lower signal to noise ratio (Eb/No). For example, according to this graph, if we aim for $P e = 1 e - 5$ , then if we use BPSK we will require an Eb/No of about 10, whereas if using BFSK or BASK we would require an Eb/No which is double that, of about 20.

SEDSAT-2 Communications Design Notes 20071125

From SEDSWiki

Contents

Modulation Schemes

Introduction

Comparing modulation schemes

Digital modulation

Binary Amplitude Shift Keying (BASK)

Binary Frequency Shift Keying (BFSK)

Binary Phase Shift Keying (BPSK)

Results & Summary

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