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

LTI system example: RC low-pass filter

Linear time-invariant (LTI) systems can be represented by the transfer function. It determines the output signal of an LTI system for a given input signal in the frequency domain.

Transfer function
Y(f)=H(f)X(f) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiaacI cacaWGMbGaaiykaiabg2da9iaadIeacaGGOaGaamOzaiaacMcacqGH flY1caWGybGaaiikaiaadAgacaGGPaaaaa@429B@
LTI system's output is related to the input by the transfer function H(f) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiaacI cacaWGMbGaaiykaaaa@3908@ .

In this briefing and the subsequent experiment an RC low-pass filter serves as example for an LTI system. The transfer function, amplitude response and phase response are derived. An RC low-pass filter is a potential divider circuit containing a resistor and a capacitor. It implements a first order low-pass.

RC lowpass   Transfer function of RC lowpass

Recalling the capacitor impedance the transfer function results to

H(f)= 1 j2πC R+ 1 j2πC = 1 1+j2πRC MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiaacI cacaWGMbGaaiykaiabg2da9maalaaabaWaaSGaaeaacaaIXaaabaGa amOAaiaaikdacqaHapaCcaWGdbaaaaqaaiaadkfacqGHRaWkdaWcca qaaiaaigdaaeaacaWGQbGaaGOmaiabec8aWjaadoeaaaaaaiabg2da 9maalaaabaGaaGymaaqaaiaaigdacqGHRaWkcaWGQbGaaGOmaiabec 8aWjaadkfacaWGdbaaaaaa@4E46@
or
H(f)= 1 1+j(f/ f C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiaacI cacaWGMbGaaiykaiabg2da9maalaaabaGaaGymaaqaaiaaigdacqGH RaWkcaWGQbGaaiikamaalyaabaGaamOzaaqaaiaadAgadaWgaaWcba Gaam4qaaqabaGccaGGPaaaaaaaaaa@41A8@ where f C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGdbaabeaaaaa@37D5@ is the cutoff frequency f C = 1 2πRC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGdbaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikda cqaHapaCcaWGsbGaam4qaaaaaaa@3DC8@ .

The rearrangement in polar coordinates leads to:

H(f)= 1 1+(f/ f C ) 2 e jarctan(f/ f C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiaacI cacaWGMbGaaiykaiabg2da9maalaaabaGaaGymaaqaamaakaaabaGa aGymaiabgUcaRiaacIcadaWcgaqaaiaadAgaaeaacaWGMbWaaSbaaS qaaiaadoeaaeqaaOGaaiykamaaCaaaleqabaGaaGOmaaaaaaaabeaa aaGccqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGQbGaeyyXIC TaciyyaiaackhacaGGJbGaaiiDaiaacggacaGGUbGaaiikamaalyaa baGaamOzaaqaaiaadAgadaWgaaadbaGaam4qaaqabaWccaGGPaaaaa aaaaa@531B@

Thus the absolute value | H(f) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaaca WGibGaaiikaiaadAgacaGGPaaacaGLhWUaayjcSdaaaa@3C29@ yields the amplitude response, and the argument arg(H(f)) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyyaiaack hacaGGNbGaaiikaiaadIeacaGGOaGaamOzaiaacMcacaGGPaaaaa@3D28@ yields the phase response.

| H(f) |= 1 1+(f/ f C ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaaca WGibGaaiikaiaadAgacaGGPaaacaGLhWUaayjcSdGaeyypa0ZaaSaa aeaacaaIXaaabaWaaOaaaeaacaaIXaGaey4kaSIaaiikamaalyaaba GaamOzaaqaaiaadAgadaWgaaWcbaGaam4qaaqabaGccaGGPaWaaWba aSqabeaacaaIYaaaaaaaaeqaaaaaaaa@44D4@

φ(f)=arctan(f/ f C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOMaai ikaiaadAgacaGGPaGaeyypa0JaeyOeI0IaciyyaiaackhacaGGJbGa aiiDaiaacggacaGGUbGaaiikamaalyaabaGaamOzaaqaaiaadAgada WgaaWcbaGaam4qaaqabaGccaGGPaaaaaaa@45C1@

Amplitude response |H(f)| of RC low-pass with fc = 10MHz
Amplitude response |H(f)| of RC low-pass with fc = 10MHz

Calculate the missing values in the table!

f (frequency) [MHz] 5 10 15 20 40
y ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaaja aaaa@3705@ (output amplitude) [V]             0.707             0.447            
φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOgaaa@37B4@ (phase shift) [ ° MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiSaalaaa@37E3@ ] 45.00 63.44
Example output amplitude and phase shift for fc =10MHz and x ^ =1V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaja Gaeyypa0JaaGymaiaadAfaaaa@39A0@

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