Periodic & Non-periodic Analog Signal

1) A composite signal is composed of three sine waves. The first wave has a frequency of 1 Hz, phase shift of 0 and maximum amplitude of 5 volts; the second wave has a frequency of 1 Hz, phase shift of 180 degrees (π radians) and maximum amplitude of 2 volts, and the third wave has a frequency of 2 Hz, phase shift of 0 and maximum amplitude of 1 volt.

a) What is the bandwidth of this composite signal? Is this a periodic or non-periodic signal?

b) Calculate the value of the voltage of this composite signal at each of the following two points of time:

         i) t = 0.25 second and ii) t = 0.75 second.

c) Draw a time domain plot in the interval t= 0 to t= 1 second for the three sine waves, then plot the approximate shape of the composite signal in this interval.

2) A periodic composite signal contains sine harmonics with frequencies from 10 KHz to 80 KHz. The fundamental frequency of the first harmonic of this composite signal is 10 KHz and all harmonics in the range 10-80 KHz are included in the composite signal. The signal is transmitted over a channel that has attenuation of -∞ dB for all frequencies less than or equal to 40 KHz and attenuation of -3 dB for all frequencies larger than 40 KHz. All harmonics have a maximum power of 10 milliwatts (mW) at the sender's side of the channel.

a) What is the bandwidth of the signal at the sender's side of the channel?

b) How many harmonics reach the receiver?

a) What is the bandwidth of the signal at the receiver's side of the channel?

c) What is the maximum power (in mW) of each received harmonic?

3) Consider a channel between sender S and receiver R.

a) The channel has a bandwidth of 20 KHz and a signal-to-noise ratio SNR = ∞. Calculate the bit rate of this channel using a digital signal that has four voltage levels.

b) The channel has a bandwidth of 8 KHz and a signal-to-noise ratio SNR = 63. Calculate the theoretical highest bit rate (i.e., capacity) of this channel.

c) Sender S transmits fixed length packets to receiver R over a link at the speed of 1000 packets/second. The one-way propagation delay of this link is 0.01 second. If the cable of the link is cut, all packets in transit on the link are lost. What are the possible values of the number of lost packets when the cable is cut?

4) A non-periodic analog signal has a spectrum with frequencies between 0 and 5000 Hz. The signal is sampled at the Nyquist sampling rate that guarantees the successful reconstruction of the analog signal. At each sample point, the intensity of the signal is measured and is encoded as an integer number of length 8 bits.

a) What is the bit rate of the traffic generated by the digitized signal?

b) Assume the digitized signal is transmitted as a stream of packets from the sender to the receiver over two hops (links) each of which has an attenuation of -1.5 dB. The power of transmission at the sender's side is 50 milliwatts. What is the power of the signal at the receiver's side?

c) Assume the one-way propagation delay for the first link in part (b) is 100 milliseconds.

Using the transmission speed (bit rate) obtained in part a, what is the value of the traffic volume (in bits) that fills the first link?

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