Question: What is the total delay (latency) for a frame of size 5 million bits that is being sent on a link with 10 routers each having a queuing time of 2 μs and a processing time of 1 μs. The length of the link is 2000 Km. The speed of light inside the link is 2 × 108 m/s. The link has a bandwidth of 5 Mbps. Which component of the total delay is dominant? Which one is negligible?
Distance (length of the link) =2000 Km
Frame size=5million bits
106 bps (1Mbps=106 bps)
2 (10 routers each having a queuing time of 2)
1 (10 routers each having a processing time of 1)
Total delay (Latency) = Propagation time + Transmission time + Queuing time
+ Processing delay.
Therefore, Propagation time=0.01s
Therefore, Transmission time=1s
Therefore, Queuing time=0.00002s
Therefore, Processing delay=0.000010s
Therefore, Total delay (Latency) =0.01+1+0.000020+0.000010
Question: In many practical situations a signal is recorded in the presence of an echo, which we would like to remove by appropriate processing. For example, in Figure P7.41 (a), we illustrate a system in which a receiver simultaneously receives a signal x(t) and an echo represented by an attenuated delayed replication of x(t). Thus, the receiver output is
Assume that x(t) is band limited [i.e., X(jω) = 0 for
and that | α | < 1.
, and the sampling period is taken to be equal to determine the difference equation for the digital filter hire] such that yc(t) is proportional to x(t).
(b) With the assumptions of part (a), specify the gain A of the ideal lowpass filter such that
(c) Now suppose that
. Determine a choice for the sampling period T, the lowpass filter gain A, and the frequency response for the digital filter h[n] such that is proportional to x(t).
(a) The received signal is,
It is ideally sampled with an impulse train of period,
Sampling with impulse functions is ideal sampling.
Translate the received signal to frequency domain.
The sampled signal in time domain is (or) the impulse train is,
It is a sequence of impulses with amplitudes equal to the sample value.
Remove the sampling period, as they simply correspond to a sample.
Convert the impulse train to a sequence.
Therefore, the sequence is,
The final output must be proportional only to
That implies the signal
must be a baseband signal and must be proportional to .
For the final output to be proportional to be only
; the frequency domain translation of the received signal is,
That means in the delayed version of
, needs to be eliminated, to get output exactly proportional to .
In frequency domain, this translates to elimination of the
So the digital filter
must eliminate term.
is applied to the digital filter, .
The expression for output
Convolution in time domain translates to multiplication in frequency domain.
Substitute for (since the output is proportional to input in time domain), for .
Thus, the frequency domain description of the digital filter is,
Determine the difference equation.
Apply inverse Fourier transform in equation (1).
Thus, the difference equation of the filter is
The output sequence
is converted to an impulse train.
That is, the sequence itself is an impulse train.
To recover the signal form this impulse train, the signal is passed through a low pass filter.
The output of the digital filter is taken to be exactly equal to.
Thus, the input to the low pass filter is
The low pass filter simply acts as a reconstruction filter.
The input is
, for the output of the ideal low pass filter to be exactly ,
Thus, the pass band gain of the ideal low pass filter is
Note that in this case,
The minimum sampling frequency, to avoid aliasing is,
The delay of the echo has a range of,
The sampling frequency of the sum of two signals is twice the bandwidth of the signal with the highest bandwidth.
The discrete sequence is,
The sampling frequency does not change if the signal is shifted.
The received signal is the sum of the signal
and the its shifted version .
The sampling frequency of sum of two signals is the twice the bandwidth of the signal with the highest bandwidth.
Thus, the sampling frequency is,
Thus, the sampling period must be ,
Apply Fourier transform to equation (1).
Determine the frequency response of the digital filter,
For the final output to be proportional to,
The only possibility is altering the function using
The input signal is,
Since the input is directly applied to the digital filter,
The frequency response of the output is multiplication of
for and for .
Thus, the gain of frequency response of the digital filter is,
The impulse train
is a sampled version of the signal ,due to the effect of digital filter.
For the final output to be exactly
, we need to pass it through a low pass filter.
The input to the ideal low pass filter is a sampled version of