在带限信道中传输方波的MatlabSimulink仿真

Simulation of Transmitting Square Waveform Signalsthrough Bandwidth Limited Channel UsingMatlab/SimulinkSun PengSchool of Telecommunication Engineering ，Beijing University of Posts and Telecommunications,Beijing (100876)E-mail ：gauss.1982@AbstractTheory of transmitting a sequence of square signal through bandwidth limited channel is given. The process is simulated in Matlab/Simulink with clearly-stated pre-assumptions. Simulation results are present to show that reconstructed signal after the square wave passing through the channel will distort and some frequency limitation must be satisfied to avoid inter symbol interference.Keywords ：bandwidth limited channel ； simulation ；Matlab ；Simulink1. IntroductionAll the information, like speech, audio, video, text, etc, can be discredited as bit stream of “0” and “1”. In real world communication we must send such “0” “1” stream modulated by square waveform signals through bandwidth channel. However, we can not transmit the signal as quick as we desire, because of some physical restriction. This thesis will discuss this topic and simulate it.2. TheoryConsider an ideal square signal: 1, -/2/2 ()0,T t T f t otherwise <<⎧=⎨⎩(1) We can obtain its spectrum via Fourier Transform [1]: sin()()T Tf F f Tfππ= (2) Equation (2) is also called Sampling Function. It is well known that time and frequency restricts each other (also known as Uncertainty Principle). So in (2) if only the lower frequencies are reserved and higher frequencies are abandoned, the signal in time domain will on longer localize in time range [-T/2, T/2]. The signal will “leak” to other time interval.Usually a stream of square signals is sent to represent bit “0” or “1” that carries information. The stream transcends a band limited channel so that higher frequencies are lost. The leak of each square signal will affect the others, to the effect that the decision module, sampling the continuous signal periodically, may wrongly judge “0” as “1” or vice versa due to the distortion of signal amplitude. Or, inter symbol interference occurs.We must point out that the discussion above is rather intuitive. However it is sufficient to understand what happens when square waveform stream transcends bandwidth limited channel and what is inter symbol interference. A model approximates band limited transmission system better is given as.And the so-called Nyquist Law [2] can be derived based on the model. It says that inter symbol free transmission can be achieved provided that(s m sm X f T T ∞=−∞+=∑ (3)Where Ts is the time period of the signal stream, X is the spectrum characteristic function of entire transmission system. Assume the cut off frequency of channel is W.1.If Ts <1/(2W), inter symbol interference is doomed to occur.2.If Ts=1/(2W), x(t), the Inverse Fourier Transform of X(f), must be of the shape SamplingFunction to avoid inter symbol interference. Unfortunately, Sampling Function has infinite signal of the entire time interval so that it can not be implemented physically.3.If Ts>1/(2W), we can eliminate inter symbol interference by tune x(t) appropriately to fulfill (3).For example, x(t) of the shape Raised Cosine function.From above 3 items[2], we understand that signals can not be transmitted too quickly, or inter symbol interference may occur.3. SimulationBased on above discussion we design a rather simplified model of band limited transmission system. Simulation is implemented using Simulink. Results are given to answer:1.What would the reconstructed signal look like after the signal was passed through the channel?2.What range of frequencies can we transmit before we start to observe inter symbol interference?3.1 The ModelDiagram[3][4] is as Fig 1.Fig 1 Simulation diagramIn this model such theoretical assumptions are made:1.Square waveform represents bit “1”, otherwise bit “0”. That is, we don’t use complicatedmodulation method.2.No transmitter and receiver filters are included.3.No Noise is added.4.The decision module is omitted.It is helpful to understand the problem although it is highly simplified.We use a Repeating Sequence from Simulink/Sources Library to represent Source. The parameters are set as Fig 2.Fig 2 Parameters of source blockFig 3 Source waveformThe signal in time interval [0, 1] looks like Fig 3. It represents 5 bit “1 0 1 0 0” where bit “1” is a square waveform with time width 0.2s. The bit pattern repeats in other interval [1,2], [2,3]… So during the whole simulation, this block will send out repeated square waveform that well approximates the statistic characteristic of Source. Note that Ts, the time width of square pulse, equals to 0.2s.The band width limited channel is simulated by a Low-pass Filter, from Signal Processing Blockset/Filtering/Filters Design Library. The parameters are as Fig 4.Fig 4 Parameters of channel block (Butterworth filter)A Butterworth Low-pass Filter is used. Its order is set as 10 to acquire sharp edge of pass band. The “Passband edge frequency” can be adjusted to obtain different channel characteristics. Note the unit of this parameter is rad, so a factor of 2*pi is necessary when 20Hz is desired, as Fig 4.3.2 ResultsBased on Section 1, we can discuss the relationship between Ts, the time width of square waveform, and W, the pass band edge frequency as followed. Les’s set Ts = 0.2s which is invariant and vary W. 1.W > 1/(2*Ts). Ts = 0.2s, 1/(2*Ts) = 2.5Hz. We set W = 20Hz. Pink signal is ideal squarewaveform while yellow signal is distorted due to band limited channel. But no inter symbol interference is observed. See Fig 5.Fig 5 Reconstructed signal (yellow) vs. original signal (pink), when W > 1/(2*Ts)2.W = 1/(2*Ts). We set W = 2.5 Hz. Things become subtle. See Fig 6. Note the “received signal” isdelayed with regard to “transmitted signal” (the delay is introduced by low pass filter, has no relation to inter symbol interference), but it is still possible to discern the “1 0 1 0 0 ” pattern. That is, there is already light amount inter symbol interference.Fig 6 Reconstructed signal (yellow) vs. original signal (pink), when W = 1/(2*Ts)3.W < 1/(2*Ts). We set W = 1Hz. See Fig 7. We can observe heavy inter symbol interference. The“1 0 1 0 0 ” pattern can not be recognized!Fig 7 Reconstructed signal (yellow) vs. original signal (pink), when W < 1/(2*Ts)4. Conclusion and Future WorksBand limited communication channel is explained and simulated. Thus Nyquist law is illustrated. Future works include modeling diagram much closer to real world (For example: modulation channel instead of base band one, introduction of Additive Gaussian White Noise, etc.).References[1]V. Oppenheim et al (1996). Signal and System (2nd Edition). New Jersey: Prentice hall[2]J. Proakis (2000). Digital Communications (4th Edition). New York: McGraw-Hill[3]The Mathworks(2007) Matlab Users’ Guide. Massachusetts: The Mathworks[4]he Mathworks(2007) Simulink Users’ Guide. Massachusetts: The MathworksAuthor Brief IntroductionSun Peng received the BS degree in Communication Engineering from Wuhan University of Technology，in 2005. He is now pursuing the MS degree in Communication and Information System in Beijing University of Posts and Telecommunications.。