# Peak-to-average power ratio of OFDM systems

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# Peak-to-average power ratio (PAPR) of OFDM systems

This tutorial points out that the peak power of an OFDM system is N times the average power - where N is the number of subcarriers. Thus, the PAPR of an OFDM system is N.

 $PAPR= | s peak | 2 S S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadg eacaWGqbGaamOuaiabg2da9maalaaabaGaaiiFaiaadohadaWgaaWc baGaamiCaiaadwgacaWGHbGaam4AaaqabaGccaGG8bWaaWbaaSqabe aacaaIYaaaaaGcbaGaam4uamaaBaaaleaacaWGtbaabeaaaaaaaa@4406@$
PAPR is the peak power divided by the average power.
 $PAPR= | s peak | 2 S S = N S S S S =N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadg eacaWGqbGaamOuaiabg2da9maalaaabaGaaiiFaiaadohadaWgaaWc baGaamiCaiaadwgacaWGHbGaam4AaaqabaGccaGG8bWaaWbaaSqabe aacaaIYaaaaaGcbaGaam4uamaaBaaaleaacaWGtbaabeaaaaGccqGH 9aqpdaWcaaqaaiaad6eacaWGtbWaaSbaaSqaaiaadofaaeqaaaGcba Gaam4uamaaBaaaleaacaWGtbaabeaaaaGccqGH9aqpcaWGobaaaa@4B9E@$
PAPR of an OFDM system is N - the number of subcarriers.

Let's compare single carrier and OFDM systems. Initially both shall use QPSK modulation and transmit with a power of $2 V 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaayk W7caWGwbWaaWbaaSqabeaacaaIYaaaaaaa@3A02@$. The difference is that the OFDM transmission signal is the output of an IFFT that is power invariant normalized. The point is that the IFFT transforms some sets of QPSK symbols, i.e. OFDM frequency domain symbols, to OFDM time symbols where all signals disappear but one, see examples in table. To keep the average power the peak power of the remaining signal is N times the average power.

Single Carrier QPSK symbols Signals of OFDM symbol
Equal QPSK symbols [1, -1]
OFDM time symbol where all signals disappear but one
Alternating QPSK symbols [1, -1], [-1, 1]
Another OFDM time symbol where all signals disappear but one
When OFDM subcarriers add up coherently the power of the transmitted signal is N times the average power. The Peak-to-average power ratio (PAPR) of is the number of subcarriers. In this example, all amplitudes and the peak amplitude of QPSK are $2 V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIYaaaleqaaOGaamOvaaaa@37B3@$. Peak amplitude of OFDM is $8+8 V=4 V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI4aGaey4kaSIaaGioaiaaykW7aSqabaGccaWGwbGaeyypa0JaaGin aiaaykW7caWGwbaaaa@3F12@$ (N = 8).

This applies to OFDM systems in general:

• OFDM adds a on top of any preexisting PAPR.
• M-QAM: For any constellation size OFDM adds a on top of the PAPR of M-QAM.
(N peak power QAM symbols may result in an OFDM signal with N times the peak power.)
• Quadrature modulation contributes 3 dB PAPR.
• The guard interval doesn't change the average and peak power and PAPR.
• Unused (virtual) subcarriers don't count. N is the number of used subcarriers. See experiment
• Variation of the transmit power doesn't change the PAPR.

In this experiment the Peak-to-average power ratio of OFDM systems (PAPR) is analyzed. As pointed out in the tutorial, the PAPR of an OFDM system is N, the number of subcarriers.

 $PAPR=N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadg eacaWGqbGaamOuaiabg2da9iaad6eaaaa@3B17@$
OFDM symbols are generated by an IFFT with size $N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@36CA@$.

## Start

The simulation starts with this setup.

 Number of subcarriers, FFT size $N=8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2 da9iaaiIdaaaa@3892@$ Modulation 4-QAM Transmission power $1 V 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaayk W7caWGwbWaaWbaaSqabeaacaaIYaaaaaaa@3A01@$

## Experiment

Determine the constellation points with peak power and calculate the PAPR of this OFDM system!

OFDM transmitted signal constellation, $N=8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2 da9iaaiIdaaaa@3892@$.

## Next steps

• Now 6 subcarriers shall be used (FFT length = 8, Virtual subcarriers = 2). Determine the PAPR.
• Check if the guard interval has impact on the PAPR. Vary the prefix length.
• Change the constellation size and analyze its impact on PAPR.
• Analyze the RF-signal transmitted by a quadrature modem (Modem = Quadrature.) Determine the PAPR. Use the multimeter function Peak-to-average power ratio.
• Vary the number of subcarriers (FFT length) and determine the PAPR. Patience required for FFT length 16 or greater.