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John Anthes, RF Monolithics, Dallas, Texas
OOK, ASK and FSK Modulation in the Presence of an
Interfering signal
This paper discusses three popular modulation schemes in the
presence of an interfering signal. For the purpose of our
discussion OOK modulation (On/Off Key) is the special case of
ASK (Amplitude Shift Key) modulation where no carrier is
present during the transmission of a zero. FSK modulation
(Frequency Shift Key) is commonly believed to perform better in
the presence of interfering signals. However, it is usually more
difficult and expensive to implement. This paper will review the
three modulation types and develop a mathematical model for the
prediction of error due to interference. This is then followed by
laboratory test results.
OOK modulation is a very popular modulation used in control applications. This is in part due to its simplicity and low implementation costs. OOK modulation has the advantage of allowing the transmitter to idle during the transmission of a “zero”, therefore conserving power. The disadvantage of OOK modulation arises in the presence of an undesired signal. As the proliferation of control and data communication apparatus increases, so does the aggravation of not being able to communicate. The three modulation types can be graphically represented on a two dimensional ortho-normal plot, sometimes referred to as a signal diagram. Consider a set of two basis vectors ϕ 1 and ϕ 2. The signal diagrams for OOK, ASK and FSK are shown in figures 1, 2 and 3 respectively. Each diagram represents the additive noise with a dashed line circle around the signal.
Figure 1
ASK
Figure 2
ϕ (^2)
ϕ (^1)
ϕ (^2)
ϕ (^1)
Threshold
Threshold
OOK
PROBABILITY OF ERROR FOR OOK AND FSK
There is an abundance of information available in literature discussing the probability of error for each of these modulation types^1. In the case of OOK modulation, the probability density of the envelope, when no signal is present, is the Rayleigh distribution. When a signal is present, the density is Rician. The total probability of error is determined by two possible error conditions. The probability of a “1” being sent and the receiver mistaking it for a “0” (a miss) and the probability of a “0” being sent and the receiver detecting a “1” (false alarm). The total probability of error is defined in the formula below:
(^1). 2 0 d
Vt p1( )r r^12. d Vt
∞ p0( )r r (1)
The variable Vt is the threshold where the two probability density functions intersect. The functions p1 and p0 are the Rician probability density function and the Rayleigh density function respectively. The variable r is the random variable of the envelope detector’s output. It is assumed that the likelihood that a “1” or a “0” is transmitted is 1/2. The evaluation of the above formula yields the approximate probability of error for OOK modulation using envelope detection:
P εaskSNR 12. 1.^1. π (^) SNR
exp SNR 4 (2)
The SNR variable in the above formula is defined as the signal to noise ratio. Likewise, the probability of error can be calculated for a receiver system using noncoherent FSK modulation. The system is modeled with two matched filters centered at f 1 and f 2 with envelope detectors summed to a decision circuit. The following expression yields the probability of error for noncoherent FSK modulation.
P εfskSNR 12. exp
SNR (^4) (3)
Pε =
The plot below shows both functions plotted together. The conclusion is that there is very little difference in the probability of error between OOK modulation and noncoherent FSK modulation.
1 10 (^0 5 10 15 ) 12
1 10 11
1 10 10
1 10 9
1 10 8
1 10 7
1 10 6
1 10 5
1 10 4
1
Pε askSNR
Pε^ fskSNR
dBSNR
Plot 1
PROBABILITY OF ERROR WITH AN INTERFERING SIGNAL (OOK)
For the purpose of simplicity, the Gaussian distribution will be used to develop the models that follow. The Gaussian distribution is an accurate model for coherent detection methods. Coherent detection offers approximately 1 dB improvement over the noncoherent detection methods at a reasonable signal to noise ratio. The figure below illustrates the OOK modulation in the presence of an interfering signal.
The integral below determines the probability that a random variable z, with an arbitrary mean Sd 1 and variance σ^2 , is less than the threshold (Vt):
Pmiss 1.
- π
d ∞
Vt Sd 1 σ exp z z
2 2 (6)
The above expression defines the probability of a “miss”. The transmitter transmitted a “1” and the receiver declared it to be a “0”. The variable Vt, which is the threshold, has previously been stated to be between the “1” and “0” levels, or 1/2 of Sd 1. The variables σ^2 , and Sd 12 are equal to the average power of the noise and the power of the desired signal when a “1” is transmitted, respectively. Similar probability functions can be written for the case when the desired signal is a “0” and the receiver declares a “1” (False alarm);
Pfa 1.
- π
d Vt σ
∞ exp z z
2 2
(7)
and for the case when an interfering signal jams the receiver,
Pjam 1.
- π
d Vt Si 1 σ
∞ exp z z
2 2
(8)
The total probability of error, using Table 1 and the above expressions, can be stated as follows:
Perr .5 P. miss .25 P. fa .25 P. jam (^) (9)
The graph below plots equation 9 for the desired signal having signal to noise ratios of 10, 15, and 20 dB. The x axis indicates the difference between the desired signal level and the interfering signal level in dB (Desired Signal -
Interfering Signal). This graph shows how an interfering signal effects the probability of error for an OOK transmission. As the interfering signal level approaches 20 dB below the desired signal level little noticeable effect is seen on the overall probability of error. The probability of error then assumes the value as if no interfering signal is present.
1 10 0 5 10 15 20 25 30 35 40 13
1 10^12
1 10^11
1 10^10
1 10^9
1 10^8
1 10^7
1 10^6
1 10^5
1 10^4
1
Desired SNR = 10 dB Desired SNR = 15 dB Desired SNR = 20dB
Probability of error
Desired Signal - Interfering Signal (dB)
Plot 2
10 12 14 16 18 20
1 10^11
1 10^10
1 10^9
1 10^8
1 10^7
1 10^6
1 10^5
1 10^4
4.77 dB carrier delta 8.4 dB carrier delta 16 dB carrier delta
Probability of Error
Signal to Noise ratio
Plot 3
PROBABILITY OF ERROR WITH AN INTERFERING SIGNAL (ASK)
Table 2, below, summarizes the possible error conditions that exist for an ASK modulated signal with an interfering signal present. It is important to remember that the detector is an envelope detector. Therefore, the decision circuitry sees the largest event at any given time, whether the signal is the interferer or the desired signal. As previously stated, the interfering signal and the desired signal are independent and the probability of either transmitting a “1” or “0” is equal. It is also assumed that the interfering signal is OOK modulated.
Transmitted Probability Receiver Error Sd=0; Si=1 .25 Si > Sd 0 ; & Si>Vt; Receiver detects a “1” (Jammed) or Sd 0 > Si; & Sd 0 >Vt; Receiver detects a “1” (False alarm) Sd=0; Si=0 .25 Sd 0 > Vt; Receiver detects a “1” (False alarm) Sd =1; Si=0 .25 Sd 1 < Vt; Receiver detects a “0” (Miss) Sd =1; Si=1 .25 Sd 1 > Si; & Sd 1 < Vt; Receiver detects a “0” (Miss) or Si > Sd 1 ; & Si< Vt; Receiver detects a “0” (Jammed)
Table 2
P( case1 ) .25. P (^) Si> Vt givenSi> Sd 0
P (^) Sd 0 > Vt givenSd> 0 Si
P( case2 ) .25 P. Sd 0 >Vt
P( case3 ) .25 P. Sd 1 <Vt
P( case4 ) .25. P (^) Sd 1 < Vt givenSd> 1 Si^ (10)
The total probability of error is the sum of the four individual contributing errors identified above in equations 10. The following three plots show the predicted probability of error of an ASK signal with an interfering signal present. Plot 4 is an ASK signal with a delta of 4.77 dB and signal to noise ratios of 10, 15 and 20 dB. Plot 5 is an ASK signal with a delta of 8.4 dB and signal to noise ratios of 10, 15, and 20 dB. Likewise, plot 6 is an ASK signal with a delta of 16 dB. Each of these plots show the effects of the probability of error as a function of the interfering signal level. Comparisons between plots 2 and plots 4, 5, and 6 show that the effects of the interfering signal on the probability of error can be diminished by transmitting ASK versus OOK. Plots 4, 5, and 6 also show that decreasing the delta between the two carrier levels for a “1” or a “0” diminish the effect caused by the interfering signal. However, The trade-off, displayed in plot 3, is that the probability of error worsens as the delta is decreased.
8.4 dB delta ASK signal with an interfering signal
0 5 10 15
1 10^10
1 10^9
1 10^8
1 10^7
1 10^6
1 10^5
1 10^4
1
10 dB SNR 15 dB SNR 20 dB SNR
Probability of Error
Desired Signal - Interfering Signal (dB)
Plot 5
16 dB delta ASK signal with an interfering signal
0 5 10 15
1 10 11
1 10 10
1 10 9
1 10 8
1 10 7
1 10 6
1 10 5
1 10 4
1
10 dB SNR 15 dB SNR 20 dB SNR
Probability of Error
Desired Signal - Interfering Signal (dB)
Plot 6
LABORATORY RESULTS
The tests were conducted using a standard encoder/decoder pair that required 36 bits of error free data in order to indicate a valid response. The data was transmitted at 2.4 Kbps. The test set up is shown below:
The tables below indicates the level of the desired signal and the level at which the interferer degraded the receivers’ performance by approximately 50 percent.
Desired Signal Modulation - OOK Minimum Sensitivity -92 dBm Desired Signal Level (OOK) Interferer Signal Level (OOK) Delta (Desired - Interferer) dB
Desired Signal
Interfering Signal (OOK)
Step Attenuator
Step Attenuator
Summer Receiver^ Decoder
Desired Signal Modulation - ASK Delta 4.77dB Minimum Sensitivity -90 dBm Desired Signal Level (ASK) Interferer Signal Level (OOK) Delta (Desired - Interferer) dB
Desired Signal Modulation - ASK Delta 9dB Minimum Sensitivity -93 dBm Desired Signal Level (ASK) Interferer Signal Level (OOK) Delta (Desired - Interferer) dB
The above data clearly shows less interference influence with the ASK modulation compared with that of the OOK modulation. Also, as predicted, the results show better sensitivity and greater influence of the interferer as the ASK delta is increased. The lab results indicate that the ASK modulation is as good or better than the FSK modulation in performance with an interfering signal. This is contrary to intuition and theory. The trends of the actual data seem to match closely with what was predicted. However, due to the fact that the actual receiver components do not behave perfectly, anomalies can be seen in the data. Some examples of non-ideal behavior could be log-amp non-linearities, receiver saturation, and noise not being perfectly correlated as theorized. The model also does not take into account that if the interfering signal is to close in frequency with the desired signal a beat note will be present within the detection pass band of the receiver. This will cause the interfering signal to have an even stronger influence on the receiver.
CONCLUSION
ASK modulation is an alternative to OOK and FSK modulation. ASK modulation offers the advantage of being more immune to interfering signals than OOK and is easier to implement at a lower cost than FSK modulation. RFM will be offering a low cost transceiver chip that will allow for both OOK and ASK modulation in the late 1998 time frame.
ACKNOWLEDGMENTS
The author wishes to thank Harry Boling and Patrick Evans for their assistance in collecting all of the laboratory results printed within this paper.
[1] Peebles, Jr., Peyton Z., Digital Communication Systems, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1987.
