
Keywords: receiver sensitivity, spread spectrum, digital communication receiver, CDMA, WCDMA, bit energy, noise power spectral density, signaltonoise ratio, SNR, signal power


Receiver Sensitivity Equation for Spread Spectrum Systems

Abstract: The following application note provides insight into how the sensitivity of a spread spectrum application is defined and how the desired sensitivity level for digital communication receivers can be calculated. This technical paper provides a stepbystep development of the receiver sensitivity equation and concludes with numeric example, putting its mathematical definition to the test.
In a spread spectrum digital communication receiver, the relationship between the link metric E
_{b}/N
_{o} (bit energy to noise power spectral density ratio) and the RF power level to achieve the desired receiver sensitivity is derived from the standard noise factor definition, F. Resulting in the receiver sensitivity equation, this relationship is used by RF designers for CDMA, WCDMA cellular receivers and other spread spectrum systems. It allows the designer to determine the receiver parameter tradeoffs in a spread spectrum link budget for any given input signal level.
Deriving E_{b}/N_{o} Relationship From Noise Factor, F
By definition, F is the ratio of signaltonoise at the input of a device (a single stage, multiple stages, or the complete receiver) to the signaltonoise at the output of the same device (
Figure 1). Since noise varies in an unpredictable manner from one point in time to the next, taking the ratio of the meansquare signal to the meansquare noise forms the signaltonoise ratio (SNR).
Figure 1.
Following are the definitions for parameters used in Figure 1 and for the sensitivity equation:
S
_{in} = available input signal power (W)
N
_{in} = available input thermal noise power (W) = KTB
_{RF}
where:
K = Boltzmann's constant = 1.381 × 10
^{23} W/Hz/K,
T = 290K at room temperature and
B
_{RF} = RF carrier bandwidth (Hz) = chip rate for the spreadspectrum system
S
_{out} = available output signal power (W)
N
_{out} = available output noise power (W)
G = device gain (numeric)
F = device noise factor (numeric)
F is defined as follows
F = (S_{in} / N_{in}) / (S_{out} / N_{out}) = (S_{in} / N_{in}) ×(N_{out} / S_{out})
Solving for N
_{out} in terms of the input noise, N
_{in}
N_{out} = (F × N_{in}× S_{out}) / S_{in} where S_{out} = G × S_{in}
results in
N_{out} = F × N_{in} × G
The average modulating signal power is defined as S = E
_{b} / T, where E
_{b} is the energy in the bit interval in Ws and T is the bit time interval in seconds.
The relationship for the average modulating signal power for the user's data rate is calculated as follows
1 / T = user bit rate, R_{bit} in Hz, which results in S_{in} = E_{b} × R_{bit}
Based on the previous equations, the signaltonoise ratio at the output of the device in terms of E
_{b}/N
_{o} is
S_{out} / N_{out} = (S_{in} × G) / (N_{in} × G × F) =
S_{in} / (N_{in} × F) =
(E_{b} × R_{bit}) / (KTB_{RF} × F) =
(E_{b}/ KTF) × (R_{bit} / B_{RF}),
where KTF represents the noise power (N
_{o}) in a 1bit interval.
Therefore,
S_{out} / N_{out} = E_{b}/N_{o} × R_{bit} / B_{RF}
With the RF bandwidth, B
_{RF} being equal to the chip rate W in a spread spectrum system, the processing gain (PG = W/R
_{bit}) can be defined as
PG = B_{RF} / R_{bit}
Therefore, R
_{bit} / B
_{RF} = 1/PG, which results in the output signaltonoise ratio
S_{out} / N_{out} = E_{b}/N_{o} × 1 / PG.
Note: For a system that is not spread in bandwidth (that is W = R
_{bit}), the value of E
_{b}/N
_{o} is numerically equal to SNR.
Receiver Sensitivity Equation
To determine SNR for a given input signal level, solve for S
_{in} from the noise factor equation
F = (S_{in} / N_{in}) / (S_{out} / N_{out})
or F = (S_{in} / N_{in}) ×
(N_{out} / S_{out})
S_{in} = F ×
N_{in} ×
(S_{out} / N_{out})
S
_{in} can also be expressed as
S_{in} = F × KTB_{RF} × E_{b}/N_{o} × 1/PG
In a more useful logarithmic form, take 10 × log of each term yielding units of dB or dBm. With the noise figure NF (dB) = 10 × log (F), this leads to the following receiver sensitivity equation
Sin (dBm) = NF (dB) + KTB_{RF} (dBm) + E_{b}/N_{o} (dB)  PG (dB)
Numeric Example
The following example is for a spreadspectrum WCDMA cellular base station receiver. Though the receiver sensitivity equation holds true for all levels of input signal level, this example uses the maximum specified input signal power at the minimum specified sensitivity in percent of the Bit Error Rate (%BER) for a given E
_{b}/N
_{o}. Following are the conditions for this numeric example:
 The maximum specified input signal level has to meet the minimum specified system sensitivity for a 12.2kbps digital voice data rate signal at 121dBm.
 The specified BER (0.1%) can be achieved for an E_{b}/N_{o} value of 5dB for the QPSK modulated signal.
 The RF bandwidth is equal to the chip rate, which is 3.84MHz.
 KTB_{RF}(log) = 10 × log(1.381 × 10^{23} W/Hz/K × 290K × 3.84MHz × 1000mW/W) = 108.13dBm.
 For a specified user data rate of R_{bit} equal to 12.2kbps, PG is PG = R_{chip} / R_{bit} = 314.75_{numeric} or 25dB_{log}.
 Substituting these values and solving for S_{out} / N_{out} = E_{b}/N_{o} × R_{bit} / B_{RF} yields the output signaltonoise ratio as 5dB  25dB = 20dB. This shows that spread spectrum systems actually operate with negative SNR for a spread bandwidth.
To find the maximum allowable receiver noise figure, which meets the minimum specified sensitivity simply solve for NF
_{max}, using the receiver sensitivity equation.
S_{in} (dBm) = NF (dB) + KTB_{RF} (dBm) + E_{b}/N_{o} (dB)  PG (dB)
The following steps and
Figure 2 provide additional guidance to finding NF
_{max}:
Step 1: The maximum specified RF input signal at desired sensitivity is 121dBm for WCDMA.
Step 2: Subtract the E
_{b}/N
_{o} value of 5dB, which yields the maximum allowable noise level in the user bandwidth (12.2kHz) of 126dBm.
Step 3: Determine the maximum noise level in the RF carrier bandwidth by adding a processing gain of 25dB, which results in the maximum allowable noise level of 101dBm.
Step 4: Subtract the maximum allowable noise level from device input noise level resulting in NF
_{max} = 7.1dB.
Figure 2.
Note: If a more efficient detector is used in the receiver design that only requires an E
_{b}/N
_{o} value of 3dB instead of 5dB, a receiver sensitivity level of 123dBm can be obtained for the same receiver NF
_{max} of 7.1dB. On the other hand, a higher NF
_{max} of 9.1dB can still be tolerated and meet the maximum specified input signal level of 121dBm at sensitivity for the reduced E
_{b}/N
_{o} value.
Conclusion
Using the receiver sensitivity equation,
S_{in} (dBm) = NF (dB) + KTB_{RF} (dBm) + E_{b}/N_{o} (dB)  PG (dB)
derived from the noise factor definition, designers can determine the receiver parameter tradeoffs in a spread spectrum link budget for any given input signal level, which makes it particularly useful to determine system sensitivity.
References
 CDMA Systems Engineering Handbook, Jhong Sam Lee & Leonard E. Miller, Artech House Publishers, 1998.
 CDMA RF System Engineering, Samuel C. Yang, Artech House Publishers, 1998.
© Jun 28, 2002, Maxim Integrated Products, Inc.

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APP 1140: Jun 28, 2002
APPLICATION NOTE 1140,
AN1140,
AN 1140,
APP1140,
Appnote1140,
Appnote 1140
