Question 2. [4 marks] Suppose that a signal has to propagate over a distance of 300m to arrive at thereceiver and undergoes an attenuation of 0.4. The baseband signal bandwidth is 20MHz and the carrierfrequency is fc = 10GHz.1) Find a linear time invariant model hc(t) for this channel2) Find the passband channel hp(t)3) Find the baseband equivalent channel h(t)4) Find the discrete-time equivalent channel that is sampled at the Nyquist rate h[n]
Question
Question 2. [4 marks] Suppose that a signal has to propagate over a distance of 300m to arrive at thereceiver and undergoes an attenuation of 0.4. The baseband signal bandwidth is 20MHz and the carrierfrequency is fc = 10GHz.1) Find a linear time invariant model hc(t) for this channel2) Find the passband channel hp(t)3) Find the baseband equivalent channel h(t)4) Find the discrete-time equivalent channel that is sampled at the Nyquist rate h[n]
Solution
- Find a linear time invariant model hc(t) for this channel
The linear time invariant model for this channel can be represented as a simple delay and attenuation. The delay is given by the propagation distance divided by the speed of light, and the attenuation is given as 0.4. Therefore, the model can be written as:
hc(t) = 0.4 * δ(t - d/c)
where δ(t) is the Dirac delta function, d is the propagation distance (300m), and c is the speed of light (3*10^8 m/s).
- Find the passband channel hp(t)
The passband channel is the baseband channel modulated up to the carrier frequency. This can be represented as:
hp(t) = hc(t) * cos(2πfct)
where fc is the carrier frequency (10GHz).
- Find the baseband equivalent channel h(t)
The baseband equivalent channel is the passband channel demodulated down to baseband. This can be represented as:
h(t) = hp(t) * cos(2πfct)
- Find the discrete-time equivalent channel that is sampled at the Nyquist rate h[n]
The Nyquist rate for this channel is twice the baseband signal bandwidth, or 40MHz. Therefore, the discrete-time equivalent channel is obtained by sampling the baseband equivalent channel at the Nyquist rate. This can be represented as:
h[n] = h(nT)
where T is the sampling period, which is the reciprocal of the Nyquist rate (1/40MHz = 25ns).
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