1. Sensitivity
System sensitivity is defined as the available input signal level Si for a given (SNR)O
Si is called the minimum detectable signal
An expression for Si can be derived from the noise factor definition as follows
Recall that from the previous lecture Ni=kTB for maximum delivered power 1

2. Sensitivity example
What minimum input signal will give an output signal to noise ratio of 0 dB in a system has an input impedance of 50 Ω, a noise figure of 8 dB and a bandwidth of 2.1 kHz, T=290º K?
Solution:
We can use the previous equation to find Si 2

3. Sensitivity example Alternatively Si as a ratio can be written as
Note that Si is related to the input voltage according to
The input signal voltage is then found to be 3

4. Sensitivity example
What is the minimum detectable signal or noise floor of the system in the previous example for an output signal to noise ratio of 10 dB
Solution:
If we follow the same procedures as in the previous example then we have
This shows that a larger input voltage is needed at the input of the receiver to raise the SNRO to 10 dB 4

5. Sensitivity example
Consider a communications receiver with a 50 Ω input impedance, a B of 3 kHz, and a 4-dB noise figure. What will be the minimum detectable voltage level
Solution:
The noise floor of this receiver for an output signal to noise ratio of 10 dB is found to be 5

6. Sensitivity for antenna and receiver
If an antenna is considered with the receive, then the total output noise is
Where Fa is the antenna noise factor, Fr is the receiver noise factor , Ni is the available noise from the input and Na is the noises added by the receiver
The output signal to noise ratio is 6

7. Sensitivity for antenna and receiver example
A given receiver system composed from an antenna with a noise factor of Fa=100, What will be the minimum detectable signal level if the receiver noise factor is =2.5 and the SNRO of the system is 10 dB. Assume the temperature is 290º k and the system band width is 3 kHz
dSolution:
The input signal is given by the equation 7

8. Sensitivity for antenna and receiver example
Solution: 8

9. Intermodulation distortion
All communication Rx contains some degree of non linearity
This non linearity can affect either
the frequency of input signal
Change the overall network gain
The network non linearity can be described by the power series expansion
y(x) is the network output and f(x) is the network input 9

10. Intermodulation distortion
If f(x) is given by
Then y can be written as
If y is expanded then an expression similar to the shown in the next slide will be obtained 10

11. Intermodulation distortion11

12. Intermodulation distortion
The frequency spectrum corresponds to the previous equation is illustrated below
Intermodulation distortion 12

13. Gain compression
One effect of the non linearity is that the amplitude of the signal became
Normally K3 will be negative and large A2cosω2t will mask the smaller A1cosω1t
This reduces the gain because of the third-order coefficient K3
Multiple signals will result in a further reduction of the gain 13

14. Gain compression
The ratio of the gain with distortion to the idealized (linear) gain is
This is referred to as single-tone gain compression factor
An important point to mention here is the 1-dB compression point which is defined in the next slide 14

15. 1-dB compression point
The 1-dB point is defined as the point at which the power gain is down 1 dB from the ideal
Receivers must be operated below their gain compression if nonlinear region is to be avoided 15

16. Second harmonic distortion
Second harmonics will occur at the receiver because of the K2 term
The amplitude of the second harmonic will be 16

17. Intermodulation distortion ratio
The intermodulation is caused by the cubic term of y
The cubic term will create intermodulation frequencies and
if ω1 and ω2 are of approximately the same frequency, then
The terms and will be filtered out
The terms and can not be filtered out and will appear in the output as distortion 17

18. Intermodulation distortion ratio
The intermodulation ratio (IMR) is defined as the ratio of the amplitude of one intermodulation terms to the amplitude of the desired output signal 18

19. Intercept point
The intermodulation distortion (IMD) power is defined as
If the two input amplitudes are the same, then the distortion power varies as the cube of the input power
This means for every 1-dB change in input power there is a 3-dB change in the power of the intermodulation terms, in this case 19

20. Intercept point
Where the power in one signal component and kd is the scale factor
The intermodulation ratio can then be defined as
Where Pd is the intermodulation power and Po is the desired output power 20

21. Intercept point
A normalized plot of the desired output and intermodulation powers is shown blow
Power transfer characteristics, including the third–order intermodulation distortion Pd and the two tone third order intercept Pi 21

22. Intercept point
The intercept point is defined as the value of the input power for which the IMD power is equal to the output power contributed by the linear term
At the intercept point
A receiver’s intercept point is a measure of the distortion in the receiver
It is also a measure of the Rx ability to reject large-amplitude signals lie in close frequency proximity to a weak signal targeted for reception 22

23. Intercept point example
Example: If a given system has an intercept point of +20 dBm, What is the IMR for an input signal power of dBm?
Solution: 23

24. Dynamic range
The dynamic range is defined as the minimum detectable signal to the signal power that causes the distortion power to be equal o the noise floor Nf
Note that the noise floor is defined as Nf=KTB
Recall that the ideal power is given by
Also the intermodulation distortion ratio can is 24

25. Dynamic range
If the distortion referred to the input is defined as then
When Pdi is equal to the noise floor Nf,
Therefore the dynamic range DR is OR 25

26. Dynamic range example
Example: A given receiver has an intercept point of 20 dBm. What will be the dynamic range for an output signal to noise ratio of 10 dB if the noise figure of the receiver is 8 dB and the bandwidth is 2.1 kHz?
Solution:
The minimum detectable signal can be found from 26

27. Dynamic range example
The Dynamic range is given by 27