Mathematical Definitions and Derivations

Intermodulation Calculations

Non-linearities in components create unwanted or spurious products from multiple tone inputs. Of particular concern in communications are the 2nd and 3rd order products. The relationships between fundamental power, spurious level and intercept point of these products use the following definitions:

Note replace n with 2 or 3 when referring to 2nd or 3rd order calculations respectively

 

the fundamental input power (dBm)

 

the fundamental output power (dBm)

 

the nth order input intercept point (dBm)

 

the spurious level of the nth order input intermodulation product (dBm)

 

the nth order output intercept point (dBm)

 

the spurious level of the nth order output intermodulation product (dBm)

 

the nth order output rejection ratio or relative level (dBc) of the spurious products compared to the fundamental signals.

 

the nth order input rejection ratio. This is the amount the input level must be increased (dB) to raise the nth order spurious products to the same level as the input before the increase.

 2nd Order Intermodulation Calculations

The equivalent 2nd order output intercept point for two cascaded stages is given by

(See Intermodulation Derivations for more information, Acrobat PDF format )

where

is the equivalent 2nd order output intercept point

is the 2nd order output intercept point of stage 1

is the 2nd order output intercept point of stage 2

is the Gain (dB) of stage 2

The input intercept point of the cascaded stages is related to the output intercept point by the total gain:

3rd Order Intermodulation Calculations

The equivalent 3rd order output intercept point for two cascaded stages is given by

(See Intermodulation Derivations for more information )

where

is the equivalent 3rd order output intercept point (dBm)

is the 3rd order output intercept point of stage 1 (dBm)

is the 3rd order output intercept point of stage 2 (dBm)

is the Gain (dB) of stage 2

The input intercept point of the cascaded stages is related to the output intercept point by the total gain:

Filtering Effects on IPx

Normal two-tone intermodulation performance assumes that

Filtering is often introduced to the system to reduce the interfering tones relative to the desired signal. Because of the filtering, input levels to subsequent devices are reduced, and, consequently, produce less intermodulation distortion in the final output. The net effect is to create an apparent IPx that is greater than the inherent IPx of the system without the filters. The calculations that are normally used in calculating the system intermodulation performance must be modified to accommodate the effects of filtering as

Effective 2nd order intercept

Effective 3rd order intercept

where,

X = the relative attenuation of the adjacent interferer

Y = the relative attenuation of the alternate interferer.

Note 1 Cascaded intermodulation calculations compute the worst-case scenario by summing the products constructively from stage to stage. To combine in any other manner would require knowledge of the transmission phase characteristics of the components.

 

Noise Calculations

The NF (noise figure) of a system is a measure of SNR (signal-to-noise ratio) degradation as a signal passes through the system. For a constant bandwidth, the SNR at the output will always be less than the ratio at the input due to the added noise by the system. The degree of degradation depends on the equivalent noise temperature of the system and the noise temperature of the source. For instance, if the noise added by the system has the same power as the source noise then the composite noise will be 3 dB higher, and the output SNR 3 dB lower. This relationship is expressed in the general equation

where

is the equivalent noise temperature of the system ( Kelvin)

is the source noise temperature ( Kelvin)

The standard definition of NF is based on a source temperature of 290 Kelvin; hence, the common equation

In practice, however, the source temperature may be quite low as when an antenna is pointed into deep space (20 K); therefore a receiver will have a different actual noise figure when the antenna is pointed at deep space than when pointed at a tower on the horizon. The receiver noise temperature hasn't changed only the source to which it is compared. The general equation above computes the actual noise figure. The decibel version is

When two or more stages are cascaded the equivalent noise temperature ( Kelvin) can be computed as

where

is the equivalent noise temperature of the system ( Kelvin)

is the noise temperature of the nth stage ( Kelvin)

is the gain (power ratio) of the nth stage

 

Compression Calculations

(See Compression Modeling, Acrobat PDF format )

SSB LO Noise

(See Integration of SSB LO Noise, Acrobat PDF format )