Solution of ECE 342 Test 1 S12
100
is excited by the signal
100 + jf
x ( t ) = 10 cos ( 250π t ) and the response is y ( t ) = A cos ( 250π t + φ ) . Find the numerical values of A and φ .
1.
A linear, time-invariant system with frequency response H ( f ) =
y ( t ) = 10 H (125 ) cos ( 250π t +  H (125 ))
H (125 ) =
100
= 0.6247e− j 0.89606
100 + j125
y ( t ) = 6.247 cos ( 250π t − 51.34° )
2.
A system has a frequency response H ( f ) . Some numerical values of H ( f ) are tabulated below. Find the
approximate numerical values of the phase delay t d and group delay tg for the system at a frequency of 85 Hz.
f
H( f )
H( f )
80 Hz 0.78087 −0.67474 radians
85 Hz 0.76194 −0.70449 radians
90 Hz 0.74329 −0.73282 radians
td = −
tg ≅ −
 H ( fc )
−0.70449
=−
= 1.319 ms
2π fc
2π × 85
1 d
1 −0.73282 − ( −0.67474 )
 H ( f )) ≅ −
= 0.9244 ms
(
2π df
2π
90 − 80
3.
A signal with signal power of 50 mW is transmitted through a cable with a loss of 2 dB/km. If the cable is 15 km
long, what is the numerical signal power in watts of the signal when it comes out of the cable?
2 dB/km × 15 km = 30 dB
The ratio of the output power to the input power is therefore 10 −30 /10 = 0.001 and the output signal power must be
50 µ W .
4.
A 100 MHz carrier is DSB modulated by a 20 kHz sinusoid. How many phase reversals occur in one second?
There are two sign reversals per period of the 20 kHz sinusoid. Therefore there are 40,000 phase reversals per second.
5.
A message x ( t ) with bandwidth 200 Hz and signal power Sx = 0.5 is the message to be transmitted by
modulating a 20 kHz carrier of amplitude Ac = 5 .
(a)
If the modulation is USSB, what is the numerical signal power ST of the transmitted signal and between what
two numerical frequency limits does the transmitted signal's power lie (in positive frequency space)?
(
)
25
× 0.5 = 6.25 . Therefore, for USSB the transmitted signal
2
power would be half of that or ST = 3.125 and the transmitted signal would lied between 20 kHz and 20.2 kHz.
For DSB, the transmitted signal power is ST = Ac2 / 2 Sx =
(b)
If the modulation is AM with µ = 0.8 , what is the numerical signal power Pc in the carrier, what is the numerical
signal power Psb in each sideband and between what two numerical frequency limits does the transmitted signal's power
lie (in positive frequency space)?
Pc =
Psb =
Ac2 25
=
= 12.5
2
2
1 Ac2 2
25 × 0.8 2 × 0.5
×
µ Sx =
=2
2 2
4
Frequency limits are 19.8 kHz and 20.2 kHz.
6.
A DSB signal with a carrier frequency of 30 kHz that is tone modulated at 500 Hz is synchronously demodulated
using a multiplier, a local oscillator and an ideal lowpass filter. The output of the lowpass filter drives a speaker.
(a)
If the local oscillator is at 30 kHz and is in phase with the carrier (as it arrives at the receiver) and the lowpass
filter cuts off at 2 kHz what numerical frequency or frequencies will be heard coming from the speaker?
Only 500 Hz.
(b)
If the local oscillator is at 30.3 kHz and the lowpass filter cuts off at 2 kHz what numerical frequency or
frequencies will be heard coming from the speaker?
200 Hz and 800 Hz
(c)
If the local oscillator is at 29.9 kHz and the lowpass filter cuts off at 500 Hz what numerical frequency or
frequencies will be heard coming from the speaker?
Only 400 Hz
7.
An AM signal with a carrier frequency of 30 kHz that is tone modulated at 500 Hz is synchronously demodulated
using a multiplier, a local oscillator and an ideal lowpass filter. The output of the lowpass filter is highpass filtered by an
ideal highpass filter with a cutoff frequency of 1 Hz (signals below 1 Hz are blocked) and the output of the highpass filter
drives a speaker.
(a)
If the local oscillator is at 30 kHz and is in phase with the carrier (as it arrives at the receiver) and the lowpass
filter cuts off at 2 kHz what numerical frequency or frequencies will be heard coming from the speaker?
Only 500 Hz.
(b)
If the local oscillator is at 30.5 kHz and the lowpass filter cuts off at 2 kHz what numerical frequency or
frequencies will be heard coming from the speaker?
500 Hz and 1000 Hz
(c)
If the local oscillator is at 29.8 kHz and the lowpass filter cuts off at 500 Hz what numerical frequency or
frequencies will be heard coming from the speaker?
200 Hz and 300 Hz
Solution of ECE 342 Test 1 S12
80
is excited by the signal
80 + jf
x ( t ) = 10 cos ( 250π t ) and the response is y ( t ) = A cos ( 250π t + φ ) . Find the numerical values of A and φ .
1.
A linear, time-invariant system with frequency response H ( f ) =
y ( t ) = 10 H (125 ) cos ( 250π t +  H (125 ))
H (125 ) =
80
= 0.53905e− j1.0015
80 + j125
y ( t ) = 5.3905 cos ( 250π t − 57.38° )
2.
A system has a frequency response H ( f ) . Some numerical values of H ( f ) are tabulated below. Find the
approximate numerical values of the phase delay t d and group delay tg for the system at a frequency of 85 Hz.
f
H( f )
H( f )
80 Hz
0.70711
−0.7854 radians
85 Hz 0.68536 −0.81569 radians
90 Hz 0.66436 −0.84415 radians
td = −
tg ≅ −
 H ( fc )
−0.81569
=−
= 1.528 ms
2π fc
2π × 85
1 d
1 −0.84415 − ( −0.7854 )
 H ( f )) ≅ −
= 0.93504 ms
(
2π df
2π
90 − 80
3.
A signal with signal power of 50 mW is transmitted through a cable with a loss of 1 dB/km. If the cable is 15 km
long, what is the numerical signal power in watts of the signal when it comes out of the cable?
1 dB/km × 15 km = 15 dB
The ratio of the output power to the input power is therefore 10 −15 /10 = 0.0316 and the output signal power must
be 1.5811 mW .
4.
A 100 MHz carrier is DSB modulated by a 5 kHz sinusoid. How many phase reversals occur in one second?
There are two sign reversals per period of the 5 kHz sinusoid. Therefore there are 10,000 phase reversals per second.
5.
A message x ( t ) with bandwidth 500 Hz and signal power Sx = 0.35 is the message to be transmitted by
modulating a 20 kHz carrier of amplitude Ac = 3 .
(a)
If the modulation is USSB, what is the numerical signal power ST of the transmitted signal and between what
two numerical frequency limits does the transmitted signal's power lie (in positive frequency space)?
(
)
9
× 0.35 = 1.575 . Therefore, for USSB the transmitted signal
2
power would be half of that or ST = 0.7875 and the transmitted signal would lied between 20 kHz and 20.5 kHz.
For DSB, the transmitted signal power is ST = Ac2 / 2 Sx =
(b)
If the modulation is AM with µ = 0.8 , what is the numerical signal power Pc in the carrier, what is the numerical
signal power Psb in each sideband and between what two numerical frequency limits does the transmitted signal's power
lie (in positive frequency space)?
Pc =
Psb =
Ac2 9
= = 4.5
2 2
1 Ac2 2
9 × 0.8 2 × 0.35
×
µ Sx =
= 0.504
2 2
4
Frequency limits are 19.5 kHz and 20.5 kHz.
6.
A DSB signal with a carrier frequency of 30 kHz that is tone modulated at 400 Hz is synchronously demodulated
using a multiplier, a local oscillator and an ideal lowpass filter. The output of the lowpass filter drives a speaker.
(a)
If the local oscillator is at 30 kHz and is in phase with the carrier (as it arrives at the receiver) and the lowpass
filter cuts off at 1 kHz what numerical frequency or frequencies will be heard coming from the speaker?
Only 400 Hz.
(b)
If the local oscillator is at 30.3 kHz and the lowpass filter cuts off at 1 kHz what numerical frequency or
frequencies will be heard coming from the speaker?
100 Hz and 700 Hz
(c)
If the local oscillator is at 29.9 kHz and the lowpass filter cuts off at 400 Hz what numerical frequency or
frequencies will be heard coming from the speaker?
Only 300 Hz
7.
An AM signal with a carrier frequency of 30 kHz that is tone modulated at 400 Hz is synchronously demodulated
using a multiplier, a local oscillator and an ideal lowpass filter. The output of the lowpass filter is highpass filtered by an
ideal highpass filter with a cutoff frequency of 1 Hz (signals below 1 Hz are blocked) and the output of the highpass filter
drives a speaker.
(a)
If the local oscillator is at 30 kHz and is in phase with the carrier (as it arrives at the receiver) and the lowpass
filter cuts off at 1 kHz what numerical frequency or frequencies will be heard coming from the speaker?
Only 400 Hz.
(b)
If the local oscillator is at 30.4 kHz and the lowpass filter cuts off at 1 kHz what numerical frequency or
frequencies will be heard coming from the speaker?
400 Hz and 800 Hz
(c)
If the local oscillator is at 29.8 kHz and the lowpass filter cuts off at 400 Hz what numerical frequency or
frequencies will be heard coming from the speaker?
Only 200 Hz
Solution of ECE 342 Test 1 S12
120
is excited by the signal
120 + jf
x ( t ) = 10 cos ( 250π t ) and the response is y ( t ) = A cos ( 250π t + φ ) . Find the numerical values of A and φ .
1.
A linear, time-invariant system with frequency response H ( f ) =
y ( t ) = 10 H (125 ) cos ( 250π t +  H (125 ))
H (125 ) =
120
= 0.69253e− j 0.8058
120 + j125
y ( t ) = 6.9253cos ( 250π t − 46.169° )
2.
A system has a frequency response H ( f ) . Some numerical values of H ( f ) are tabulated below. Find the
approximate numerical values of the phase delay t d and group delay tg for the system at a frequency of 85 Hz.
f
H( f )
80 Hz 0.83205
H( f )
−0.588 radians
85 Hz 0.81602 −0.6163 radians
90 Hz
0.8
−0.6435 radians
td = −
tg ≅ −
 H ( fc )
−0.6163
=−
= 1.154 ms
2π fc
2π × 85
1 d
1 −0.6435 − ( −0.588 )
= 0.88331 ms
(  H ( f )) ≅ − 2π
2π df
90 − 80
3.
A signal with signal power of 50 mW is transmitted through a cable with a loss of 3 dB/km. If the cable is 15 km
long, what is the numerical signal power in watts of the signal when it comes out of the cable?
3 dB/km × 15 km = 45 dB
The ratio of the output power to the input power is therefore 10 −45 /10 = 3.1623 × 10 −5 and the output signal power must be
1.5811 µ W .
4.
A 100 MHz carrier is DSB modulated by a 3 kHz sinusoid. How many phase reversals occur in one second?
There are two sign reversals per period of the 3 kHz sinusoid. Therefore there are 6,000 phase reversals per second.
5.
A message x ( t ) with bandwidth 500 Hz and signal power Sx = 0.85 is the message to be transmitted by
modulating a 20 kHz carrier of amplitude Ac = 7 .
(a)
If the modulation is USSB, what is the numerical signal power ST of the transmitted signal and between what
two numerical frequency limits does the transmitted signal's power lie (in positive frequency space)?
(
)
49
× 0.85 = 20.825 . Therefore, for USSB the transmitted
2
signal power would be half of that or ST = 10.4125 and the transmitted signal would lied between 20 kHz and 20.5 kHz.
For DSB, the transmitted signal power is ST = Ac2 / 2 Sx =
(b)
If the modulation is AM with µ = 0.8 , what is the numerical signal power Pc in the carrier, what is the numerical
signal power Psb in each sideband and between what two numerical frequency limits does the transmitted signal's power
lie (in positive frequency space)?
Pc =
Psb =
Ac2 49
=
= 24.5
2
2
1 Ac2 2
49 × 0.8 2 × 0.85
×
µ Sx =
= 6.664
2 2
4
Frequency limits are 19.5 kHz and 20.5 kHz.
6.
A DSB signal with a carrier frequency of 30 kHz that is tone modulated at 600 Hz is synchronously demodulated
using a multiplier, a local oscillator and an ideal lowpass filter. The output of the lowpass filter drives a speaker.
(a)
If the local oscillator is at 30 kHz and is in phase with the carrier (as it arrives at the receiver) and the lowpass
filter cuts off at 2 kHz what numerical frequency or frequencies will be heard coming from the speaker?
Only 600 Hz.
(b)
If the local oscillator is at 30.3 kHz and the lowpass filter cuts off at 2 kHz what numerical frequency or
frequencies will be heard coming from the speaker?
300 Hz and 900 Hz
(c)
If the local oscillator is at 29.9 kHz and the lowpass filter cuts off at 600 Hz what numerical frequency or
frequencies will be heard coming from the speaker?
Only 500 Hz
7.
An AM signal with a carrier frequency of 30 kHz that is tone modulated at 600 Hz is synchronously demodulated
using a multiplier, a local oscillator and an ideal lowpass filter. The output of the lowpass filter is highpass filtered by an
ideal highpass filter with a cutoff frequency of 1 Hz (signals below 1 Hz are blocked) and the output of the highpass filter
drives a speaker.
(a)
If the local oscillator is at 30 kHz and is in phase with the carrier (as it arrives at the receiver) and the lowpass
filter cuts off at 2 kHz what numerical frequency or frequencies will be heard coming from the speaker?
Only 600 Hz.
(b)
If the local oscillator is at 30.6 kHz and the lowpass filter cuts off at 2 kHz what numerical frequency or
frequencies will be heard coming from the speaker?
600 Hz and 1200 Hz
(c)
If the local oscillator is at 29.8 kHz and the lowpass filter cuts off at 600 Hz what numerical frequency or
frequencies will be heard coming from the speaker?
200 Hz and 400 Hz