通信系统习题Word文件下载.docx

1、In terms of the unit step function, this can be written as:And the energy is equal to 2T.2. Evaluate the following integrals3.Find the output of the time-invariant linear system having the impulse response h（t） shown in the Figure（a）, in response to the input signal x（t） presented in the Figure （b）.

2、 In order to calculate the convolution x（t）*h（t）, we need first to find the mirrored version of the signal x（t）, change the variable to , i.e. find x（-k）, as presentedThat mirrored version of the input signal is then shifted and signal x（t-） is considered.It is clearly visible that for t 7, x（t-） an

3、d h（t） stop overlapping for any Therefore, the output signal y（t） can be expressed as:After performing integration, we getThe obtained output signal y（t） is plotted below. Itis worth to notice that the signal y（t） is smoother than the input signal x（t）, and that the time limits property holds. 4.Det

4、ermine the auto-correlation function of the following signal （where A0）:Solution: The waveform is shown as follows Firstly note that the autocorrelation of a real function is always symmetric - thus you only need to determine the shape of the autocorrelation for r 0. Secondly, note that the time-shi

5、fted image of the triangular pulse does not overlap the original function at all if r T or r-T Therefore the autocorrelation is zero in those regions. We can just work out the autocorrelation function for 0 r 0 and 0（f） = knotherwise; k is an oddinteger.In addition, since w（t） is a real signal, 0（f）

6、 = -0（-f）.3,Find the Fourier transform of the so called radio pulses w1（t） and w2（t） defined as: The Fourier transform W,（f） can be found substituting w（t） = n（t） and 0= 0 into the modulation theorem. This substitution yields: For w2（t）, we need to take 0 = -0.5, which results in:4.Find the Fourier

7、transform of the rectangular wave w（t） shown in the figure below and draw the magnitude spectrum | W（f）| in the frequency range - 300 Hz to 300 Hz. To find the Fourier transform of the wave w（t） we will use the formulaTherefore, we need first to find the Fourier transform of the pulse h（t） which is

8、repeated at regular intervals to generate the wave w（t）. The pulse h（t） is a rectangular pulse with a width T =10 msThe pulses h（t） are repeated every T0 = 30 ms. Hence, assuming that time t is in seconds, the wave w（t） can be expressed as:and its Fourier transform W（f） is given by:where the fundame

9、ntal frequency f0 = 1/T0 = 33.333 Hz, and /-/（f） is the Fourier transform of a single pulse From the table of Fourier transforms Substituting this into the formula for W（f） yields:Because the different terms of the sum in the formula for W（f） do not overlap, we can take the magnitude of |W（f）| as eq

10、ual to the sum of the magnitudes of different terms. The weights of impulses for f = nf0; n = 0, 1, 2,can be presented in the form of a tableWeights of spectral lines for the magnitude spectrum |W（f）| Plot of the magnitude spectrum of the wave w（t） Tut51.A mixer is used to multiply two signals x（i）

11、and y（i）. Plot the magnitude spectrum of the output of the mixer ifwhere A = 2mV, fm = 2kHz, B = 1mV and fc = 200kHz. Repeat your considerations if the magnitude spectrum of x（i） is as given in Fig. 1, and y（i） is the same as previously. Solution：In the first case, the mixers output z（i） is given by

12、:Utilizing the trigonometric identity:we get:Becausethe latest can be rewritten in the formSubstituting the numerical values for the constants yields:and the magnitude spectrum of z（i） is given in Fig.2. Figure 2: Magnitude spectrum of waveform at the mixers output.To find the magnitude spectrum in

13、the second case, we need to utilize the frequency translation property of Fourier transform, which states In the considered case, 0 = 0. The magnitude spectrum of thez（i） is given in Fig.3.Figure 3: Magnitude spectrum of the z（i）.2. Draw the magnitude spectrum of the signal xout（t） at the output of

14、a non-linear device, having a characteristic:Assume that the input signal consists of a sum of a baseband signal and a carrier wave having its frequency much higher than the maximum frequency in the baseband component. Repeat your analysis, assuming that the baseband signals of the form:SolutionIn t

15、he first case, the considered input signal xin（i） is given by the equation:where m（i） is a baseband signal. Substituting this to the formula for the output voltage of the non-linear device yields:The plot of the magnitude spectrum of xoui（i） is given in Fig.4. Figure 4: Plot of the magnitude spectru

16、m at the output of a non-linear device if the input signal is a superposition of a sinusoidal wave of frequency fc and the signal m（t） having themagnitude spectrum |M（f）|.In the second part of the problem the xin（t） is of the form:Without the loss of generality, let us assume here that this correspo

17、nds to the situation where the signal m（t） considered in the previous case has a magnitude spectrum as shown in Fig.5. Figure 5: Magnitude spectrum of the signal m（t） composed oftwo sinusoids.All of the derivations performed for the first part of the problem are exactly the same in this part. Howeve

18、r, the signal m（t） is now of the form:Therefore, to find all the spectral components of the signal atthe output of the non-linear device, we need to workout thespectral components of Substituting the previous equation for a2m方（t） we get:The plot of the magnitude spectrum of the xout（t） is given in F

19、ig. 6, and the magnitudes of the spectral components are listed in Table 1. Figure 6: Magnitude spectrum of the xout（t）.Table 1: Magnitudes of spectral components for xout（t）.3. Consider a pulse w（t） given in Fig.l, and multiply it by a sinusoidIs the resulting waveform a base-band or a band-pass wa

20、veform? If the resulting waveform is a band-pass one, find its band-pass representation in terms of: the complex envelope g（t）, the real and imaginary parts of the envelope x（t） and y（t）, the magnitude of the envelope R（t） and the phase （t）. The resulting waveform is given by:To determine if it is a

21、 base-band or band-pass waveform, one needs to check its magnitude spectrum. It can be computed using the real signal frequency translation theorem Couch.Application of that theorem yields:where W（f） denotes the Fourier transform of the pulse w（t） derived in the previous example. As it is visible in Fig.3, the magnitude of Wf） decays rapidly with increase of f, and we can consider with a very good accuracy that