25-04-2011, 02:50 PM
Analog Digital Hybrid Modulation.doc (Size: 1,016.5 KB / Downloads: 93)
1. INTRODUCTION
Traditional approaches provide two primary alternatives: using analog transmission or digital transmission, of which latter uses sampling and quantization principles. Analog Modulation techniques such as FM and PM provide significant noise immunity and provide SNR improvement proportional to the square root of modulation index and thus are able to trade off bandwidth for SNR. However, the SNR improvement provided by these techniques is much lower than the ideal performance as shown by the Shannon's capacity theorem. Digital modulation techniques can use error detection codes which provide performance close to the theoretical limits. However, the quantization introduced when the analog signal is digitized introduces distortion which cannot be later recovered. If a fixed number of bits are used for quantization, a fixed signal distortion is introduced at an early stage. This distortion will be present regardless of the transmission quality of the channel being used. Thus the original signal can be considered to be permanently impaired. Communications systems are normally operate at SNR which is much higher than the minimum required, because of the need to leave a margin for fading and other effects which might occasionally reduce the SNR. So, it is essential to design a communications system where the output SNR increases as the channel SNR increases. It is an inherent property in analog modulation but not in digital modulation.
Signal Code Modulation (SCM) is a method for transmitting analog information over a noisy channel which has been developed for a commercial point-to-multipoint communication system. The technique is based on the idea of representing analog signal by a digital component and analog component. Both components are transmitted by the communication system and used to reconstruct an estimate of the signal at the receiver. Because of the SCM technique to vary the data rate according to the available SNR, it can be used for re-transmitting a digital communication signal over channels with different SNRs without complete demodulation and re-modulation (Demod/Remod). The SCM is able to trade off bandwidth for SNR, in the classical manner and achieves a performance close to that of conventional Demod/Remod methods. It does so by performing relatively simple operations on the sampled input signal and does not require detailed knowledge of its fine structure like framing, coding etc.
2. THE SCM TECHNIQUE
SCM utilizes both the analog as well as digital modulation techniques providing an analog pipe through which any band-limited waveform can pass. The primary analog input signal is sampled at the appropriate rate and quantized. The digital samples are denoted by certain symbols. However, unlike PCM, SCM does not discard the quantization error. The quantization error is extracted and transmitted over channel as an analog symbol. A.
The transmission channel is divided into two channels. Channel 1 is analog, and channel 2 is digital. In a process essentially identical to PCM, the original analog signal at the system input is sampled at the appropriate rate, based on the sampling theorem, and converted to digital values. The resulting D symbols are transmitted via channel 2 using a digital transmission technique optimized for the channel. Those D symbols represent N bits per analog input sample. To produce the quantization error A, the PCM data is converted back to analog and subtracted from the original input. This A symbol is amplified by a gain of 2N or any gain that will optimize the voltage swing of the A symbol with that of channel 1.
The SCM receiver performs the opposite operation, combining the A and D symbols into an analog stream replica of the original analog signal. This replica is not a precise copy of the original signal, because noise in the channels could vary the A symbols or cause bit errors in the D symbols. However, the 2N amplitude gain in channel 1 has provided noise power immunity of 22N to the A symbols. This is one of the key benefits of SCM and provides a near ideal scheme for error-free transmission.
Suppose we are given a band-limited signal x(t) of bandwidth B Hz, which needs to be transmitted over a channel of bandwidth Bc with Gaussian noise of spectral density N0 watts per Hz, using a transmitter with an average power of P watts. The signal is real baseband signal and that the transmission takes place in baseband. The signal is sampled at the Nyquist rate of fs = 2B samples per second, to produce a sampled signal x(n). The SCM transmission and reception processes are depicted in Figure 1.
Next, let the signal be quantized to produce a discrete amplitude signal of M=2b levels. Where b is the no. of bits per sample of the digital symbol D, which is to be encoded. More explicitly, let the values of the M = 2b levels be, q1, q2, q3, q4...qM which are distributed over the range α [-1, +1], where α is the proportionality factor determined relative to the signal. Given a sample x(n), we find the nearest level qi(n).The index i(n) in binary format is the digital representation of x(n) and qi(n) is the digital symbol and xa(n) = x(n) - qi(n) is the analog representation. The exact representation of the analog signal is given by,
x(n) = qi(n) + xa(n)
We can accomplish the transmission of this information over the noisy channel by dividing it into two channels: one for analog information and another for digital information. The analog channel bandwidth is Ba = βaB, and the digital channel bandwidth being Bd = βdB, where Ba + Bd = Bc . The channel bandwidth β = Bc / B, be the bandwidth expansion factor, i.e. the ratio of the bandwidth of the channel to the bandwidth of the signal.
Similarly, the variables βa and βd are the ratios of Ba / B and Bd / B. Here we will assume that βa = 1 so that βd = β - 1. The total transmit power is also divided between the two channels with fraction pa for the analog channel and fraction pd for the digital one, so that pa + pd = 1.
The SNR of the channels is first conveniently defined where no bandwidth expansion is used
γ = P / BN0 (1)
The SNR of the analog channel is given by
SNRa = paP / BaN0 = Pa γ / βa (2)
And the SNR of the digital channel is given by
SNRd = pdP / BdN0 = Pd γ / βd (3)
Of special interest is the case where the signal power is divided in proportion to bandwidth. This is the case where the analog and digital channels have the same spectral density of the transmitted signal. Inferring that in this case
SNRa = SNRd = γ / β (4)
The objective the communication system is to transmit the signal x(n) as accurately as possible. In other words, we want we want to design the system so as to maximize, the output SNR of the demodulated signal x”(n), where the output SNR is:
SNR0 = E {(x”(n) - x(n))2} / E{x2(n)} (5)
In the following, we calculate the output SNR as the function of the channel SNR and the bandwidth expansion factor and plot the corresponding graph.
3. MAXIMUM OUTPUT SNR
Let us consider the best possible SNR that can be obtained by bandwidth expansion, when we wish to transmit a signal of bandwidth B through a Gaussian channel of bandwidth βB, we have
SNR0 = (1 + γ / β)β – 1 (6)
Where SNR0 is the SNR at the channel output (after demodulation), γ is the channel SNR for the case where no bandwidth expansion is used and β is the bandwidth expansion factor.
It can be derived using Shannon's capacity theorem that the formula for capacity of a Gaussian Channel is given by:
C = βB log2 (1 + γ / β) (7)
While at the demodulator end, C = B log2 (1 +SNR0) (8)
The figure 2 depicts the Output SNR vs. γ for different bandwidth expansion factors