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

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

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

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

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

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

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

Sensitivity for antenna and receiver example Solution: 8

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

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

Intermodulation distortion 11

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Dynamic range example The Dynamic range is given by 27