Effect of Oscillator Phase Noise and Processing Delay in Full-Duplex OFDM Repeaters Taneli Riihonen, Pramod Mathecken, and Risto Wichman Aalto University School of Electrical Engineering, Finland Session WA4b OFDM(A), Nov. 7, 2012 46th Asilomar Conference on Signals, Systems and Computers
Introduction Taneli Riihonen Phase Noise in OFDM Repeaters 2 / 32
Problem: Coverage Gaps Coverage area of the main transmitter Shadow zone How to serve shadowed areas in cellular systems? Transmit powers cannot be increased indefinitely The transmitter density needs to be higher and non-uniform Taneli Riihonen Phase Noise in OFDM Repeaters 3 / 32
Solution: Full-Duplex Repeaters (1) S rx Capture a good quality input signal within the main coverage area highly directional receive (rx) antenna in an elevated position preferably line-of-sight to the source (S) transmitter Taneli Riihonen Phase Noise in OFDM Repeaters 4 / 32
Solution: Full-Duplex Repeaters (2) S rx tx a) D Amplify and forward the signal within the shadow zone Omnidirectional transmit (tx) antenna, e.g., for providing a) indoor coverage Taneli Riihonen Phase Noise in OFDM Repeaters 5 / 32
Solution: Full-Duplex Repeaters (3) S rx tx b) D Amplify and forward the signal within the shadow zone Omnidirectional transmit (tx) antenna, e.g., for providing b) underground coverage Taneli Riihonen Phase Noise in OFDM Repeaters 6 / 32
Solution: Full-Duplex Repeaters (4) S rx tx c) D Amplify and forward the signal within the shadow zone Omnidirectional transmit (tx) antenna, e.g., for providing c) coverage between buildings Taneli Riihonen Phase Noise in OFDM Repeaters 7 / 32
Solution: Full-Duplex Repeaters (5) S rx tx tx D D tx D Distributed tx antenna system can be also implemented Transparent coverage boost without allocating extra frequencies No wired (optical fiber) data connection needed, only power supply Taneli Riihonen Phase Noise in OFDM Repeaters 8 / 32
Problem&Solution: Self-interference Cancellation S rx tx tx D D tx D Single-frequency operation comes at the cost of self-interference The repeater s gain needs to be limited to avoid oscillation Herein: sufficient cancellation performance and gain margin Taneli Riihonen Phase Noise in OFDM Repeaters 9 / 32
Problem: Oscillator Phase Noise in OFDM BB OFDM tx multipath channel BB OFDM rx Generally speaking, orthogonal frequency-division multiplexing is robust to timing asynchronism and multipath delay spread sensitive to phase noise, carrier offset, I/Q imbalance Jumps from base band (BB) to carrier frequency f c and back to BB upconversion: a tx (t) = e j2πf ct+jθ tx (t) downconversion: a rx (t) = e j2πf ct+jθ rx (t) Focus in this work: The effect of phase noise, θ tx (t) and θ rx (t), in terms of processing delay with two different repeater designs Taneli Riihonen Phase Noise in OFDM Repeaters 10 / 32
System Model Taneli Riihonen Phase Noise in OFDM Repeaters 11 / 32
OFDM Repeater Link: Signal Model (1) X S [n] Source (S) inverse DFT CP insertion P/S and D/A a S (t) x S (t) ˆx S (t) h SR (t) ŷ R (t) Standard OFDM modulator: Frequency-domain symbols {X S [n]} N c 1 n=0 are transformed into analog baseband signal x S (t) Upconversion: Mixing x S (t) with oscillator signal a S (t) ˆx S (t) = a S (t) x S (t) where the oscillator is assumed to be ideal: a S (t) = e j2πf ct After a passband filter and a high-power amplifier, RF signal ˆx S (t) propagates to the repeater through multipath channel h SR (t) ŷ R (t) = (h SR ˆx S )(t)+ŵ R (t) Taneli Riihonen Phase Noise in OFDM Repeaters 12 / 32
OFDM Repeater Link: Signal Model (2) Repeater (R) a rx (t) β a tx (t) A/D D/A ŷ R (t) y R (t) x R (t) ˆx R (t) Downconversion: Mixing ŷ R (t) with oscillator signal a rx (t) y R (t) = a rx (t) ŷ R (t) Processing delay τ due to digital (or only analog?) filtering etc. Amplification by β, self-interference cancellation, equalization Upconversion: Mixing x R (t) with oscillator signal a tx (t) ˆx R (t) = a tx (t) x R (t) Non-ideal repeater oscillator(s): Phase noise in a rx (t) and a tx (t) Taneli Riihonen Phase Noise in OFDM Repeaters 13 / 32
OFDM Repeater Link: Signal Model (3) Destination (D) Y D [n] ˆx R (t) h RD (t) ŷ D (t) a D (t) y D (t) A/D and S/P CP removal DFT After a passband filter and a high-power amplifier, RF signal ˆx R (t) propagates to the destination through multipath channel h RD (t) ŷ D (t) = (h RD ˆx R )(t)+ŵ D (t) Downconversion: Mixing ŷ D (t) with oscillator signal a D (t) y D (t) = a D (t) ŷ D (t) where the oscillator is assumed to be ideal: a D (t) = a S (t) Standard OFDM demodulator: Analog baseband signal y D (t) is transformed to frequency-domain symbols {Y D [n]} N c 1 n=0 Taneli Riihonen Phase Noise in OFDM Repeaters 14 / 32
OFDM Repeater Link: Signal Model (4) X S [n] Y D [n] inverse DFT CP insertion P/S and D/A a S (t) h SR (t) a rx (t) β a tx (t) h RD (t) a D (t) A/D D/A A/D and S/P CP removal DFT Let us denote a R (t) = a tx (t) a rx (t τ) End-to-end baseband signal model in time domain (simplified form): y D (t) = βh RD (t) {a R (t) (h SR x S )(t τ)+w R (t τ)}+w D (t) Equivalent model in frequency domain for the nth subcarrier: Y D [n] = βh RD [n] N c 1 k=0 A R [k n](h SR [k]x S [k]+w R [k])+w D [n] Inter-carrier interference (ICI) is realized through A R [k] which corresponds to phasor a R (t) from oscillator phase noise Taneli Riihonen Phase Noise in OFDM Repeaters 15 / 32
Signal-to-Interference and Noise Ratio (SINR) Signal, interference and noise powers N c 1 E{ Y D [n] 2 } = β 2 H RD [n] 2 A R [k n] 2 ( H SR [k] 2 P S [k]+σr)+σ 2 D 2 k=0 where P S [n] = E{ X S [n] 2 }, σr 2 = E{ W R[n] 2 }, σd 2 = E{ W D[n] 2 } With sufficiently coherent channels (vs. oscillator s spectral density) N c 1 E{ Y D [n] 2 } β 2 H RD [n] 2 ( H SR [n] 2 P S [n]+σr) 2 A R [k] 2 +σd 2 k=0 Finally, the instantaneous SINR can be expressed as γ[n] = (1 α)γ SR [n]γ RD [n] (αγ SR [n]+1)γ RD [n]+ P tx[n] σ 2 R β2 where α = 1 A R [0] 2 = N c 1 k=1 A R[k] 2 represents ICI power and SNRs are γ SR [n] = P S [n] H SR [n] 2 /σ 2 R, γ RD[n] = P tx [n] H RD [n] 2 /σ 2 D Taneli Riihonen Phase Noise in OFDM Repeaters 16 / 32
Non-ideal Oscillators in the Repeater In the following: Comparison of two different repeater designs (a) Two separate oscillators (b) Reusing single oscillator conjugation a rx (t) a tx (t) a rx (t) a rx (t) amplification amplification downconversion: a rx (t) = e j(2πf ct θ rx (t)) upconversion: a tx (t) = e j(2πf ct+θ tx (t)) downconversion: a rx (t) = e j(2πf ct θ rx (t)) upconversion: a tx (t) = a rx(t) = e j(2πf ct θ rx (t)) The total phase distortion caused by phase noise and repeater processing delay τ can be captured as phasor process a R (t) = a rx (t τ) a tx (t) Taneli Riihonen Phase Noise in OFDM Repeaters 17 / 32
Wiener Phase Noise The phase noise of free-running oscillators can be modelled accurately as a Wiener process, i.e., standard Brownian motion or random walk with Gaussian steps : θ rx (t 0 ) θ rx (t 0 t) N(0,c rx t ) θ tx (t 0 ) θ tx (t 0 t) N(0,c tx t ) (In)dependence of rx and tx sides due to the repeater design Two separate oscillators: θ tx (t) is independent of θ rx (t) Reusing single oscillator: θ tx (t) = θ rx (t) The quality of the oscillator is parametrized by f 3dB which defines the 3dB bandwidth of the oscillator power spectral density (PSD) When using two oscillators, they are assumed to be of similar quality in this study: c = c rx = c tx = 4πf 3dB Taneli Riihonen Phase Noise in OFDM Repeaters 18 / 32
Spectral Spreading due to Phase Noise Taneli Riihonen Phase Noise in OFDM Repeaters 19 / 32
Example (1): Long Processing Delay (a) Two separate oscillators (b) Reusing single oscillator θ rx (t τ) θ tx (t) a R (t) 2πf c τ Taneli Riihonen Phase Noise in OFDM Repeaters 20 / 32
Example (2): Short Processing Delay (a) Two separate oscillators (b) Reusing single oscillator θ rx (t τ) θ tx (t) a R (t) 2πf c τ Taneli Riihonen Phase Noise in OFDM Repeaters 21 / 32
PSD of Repeater Phasor Process Total phase distortion caused { by the repeater e j(2πf cτ+θ rx (t τ)+θ tx (t)), two oscillators a R (t) = a rx (t τ) a tx (t) = e j(2πf cτ+θ rx (t τ) θ rx (t)), one oscillator ICI is realized through A R [k] which represents instantaneous spectral spreading for each OFDM symbol Standard steps for calculating power spectral density (PSD): first R(t) = E{a R (t 0 )a R (t 0 t)} then S(f) = 1 2π R(t)e j2πft dt PSD is related to E{ A R [k] 2 }, i.e., expected ICI power In the ideal case a R (t) = e j2πfcτ yielding S 0 (f) = δ(f) (a) Two separate oscillators S 2 (f) = 1 π c c 2 +(2πf) 2 (b) Reusing single oscillator S 1 (f) = e cτ S 0 (f)+ S 1 (f)s 2 (f) where S 1 (f) = 1 e cτ ( cos(2πfτ)+cτ sinc(2πfτ) ) Taneli Riihonen Phase Noise in OFDM Repeaters 22 / 32
Numerical Results (1) On the right: PSD when reusing single oscillator and f 3dB = 100Hz Extreme cases for processing delay: S 1 (f) { S 0 (f), when τ 0 S 2 (f), when τ (cf. OFDM sample and symbol duration) S1(f) [dbc/hz] 20 40 60 80 100 120 140 160 τ = {0.1,1,10,100, }µs 10 1 10 2 10 3 10 4 10 5 10 6 10 7 f [Hz] Except for the impulse at the zero frequency (not visible in the above figure), the PSD is approximately flat when f < 1 4τ When f > 1 4τ, the PSD decays 20dB per decade Taneli Riihonen Phase Noise in OFDM Repeaters 23 / 32
Numerical Results (2) On the right: PSD vs. processing delay when reusing single oscillator and f 3dB = 100Hz at 1kHz: Flat PSD at 1MHz: PSD decays 20 db per decade (cf. subcarrier spacing and bandwidth in OFDM) S1(f) [dbc/hz] 20 40 60 80 100 120 140 160 f = 1kHz f = 1MHz 10 8 10 7 10 6 10 5 10 4 10 3 10 2 τ [s] The PSD oscillates less when the processing delay increases and becomes smooth when τ > 1 4f 3dB ( ) However, OFDM symbols are typically shorter than that Taneli Riihonen Phase Noise in OFDM Repeaters 24 / 32
Transmission Rate Analysis Taneli Riihonen Phase Noise in OFDM Repeaters 25 / 32
Distribution of ICI Power When reusing a single oscillator, time-domain phase distortion a R (t) can be seen as colored Gaussian noise And A R [k] represents a R (t) in frequency domain after sampling Using Taylor series expansion, ICI power α = N c 1 k=1 A R[k] 2 becomes a sum of correlated gamma random variables Coefficients λ k depend on f 3dB and τ via a covariance matrix! Finally, the probability density function (PDF) of α can be expressed as a weighted sum of gamma PDFs: p(α) = κ ζ k p k (α) where p k (α) = k=0 λ N c 1 +k 2 1 Γ κ = N c 1 n=1 ζ 0 = 1 and ζ k+1 = 1/2 k+1 α N c 1 2 +k 1 e α λ 1 λ 1 λ n (probability mass normalization) ( ) iζk+1 i 1 λ1 /λ j k+1 i=1 Nc 1 j=1 ( Nc 1 2 +k ) Taneli Riihonen Phase Noise in OFDM Repeaters 26 / 32
Average Transmission Rate Repeater gain β 2 = (P tx [n]/σ 2 R )/(γ SR[n]+1) transforms SINR to γ[n] = (1 α)γ SR [n]γ RD [n] γ SR [n]+(αγ SR [n]+1)γ RD [n]+1 Instantaneous transmission rate is given by C[n] = log 2 (1+γ[n]) = log 2 ( γsr [n]γ RD [n]+γ SR [n]+γ RD [n]+1 αγ SR [n]γ RD [n]+γ SR [n]+γ RD [n]+1 Using p(α), average transmission rate can be calculated as ) γ SR [n]γ RD [n] C[n] = E{C[n]} = log 2 (1+ κ ζ k I k γ SR [n]+γ RD [n]+1 k=0 ) γ where SR [n]γ RD [n] I k = log 2 (1+ γ SR [n]+γ RD [n]+1 α p k (α)dα 0 I k can be solved in a closed form using Meijer s G-function (or generalized hypergeometric and incomplete gamma functions) ) Taneli Riihonen Phase Noise in OFDM Repeaters 27 / 32
Numerical Results (3) OFDM parameters for a DVB-T/H-like system: N c = 8192 subcarriers and 8MHz bandwidth sample duration: 0.11µs FFT duration: 896µs subcarrier spacing: 1.1kHz Oscillators: f 3dB = 100Hz C[n] [bit/s/hz] 8 7 6 5 4 3 2 1 τ = {0.1,1,10,100, }µs 0 0 5 10 15 20 25 30 γ SR [n] = γ RD [n] [db] When reusing a single oscillator, transmission rate degradation can be minimized by decreasing the processing delay Implementation with two separate oscillators means τ Taneli Riihonen Phase Noise in OFDM Repeaters 28 / 32
Numerical Results (4) OFDM parameters for a DVB-T/H-like system: N c = 8192 subcarriers and 8MHz bandwidth sample duration: 0.11µs FFT duration: 896µs subcarrier spacing: 1.1kHz Oscillators: f 3dB = 100Hz C[n] [bit/s/hz] 8 7 6 5 4 3 2 1 γ SR [n] = γ RD [n] = 25dB γ SR [n] = γ RD [n] = 20dB γ SR [n] = γ RD [n] = 15dB γ SR [n] = γ RD [n] = 10dB 0 10 8 10 7 10 6 10 5 10 4 10 3 10 2 τ [s] If the processing delay is a few tens of OFDM samples or shorter, the transmit-side noise reverts the effect of receive-side noise The delay needs to be shorter than the cyclic prefix anyway Taneli Riihonen Phase Noise in OFDM Repeaters 29 / 32
Conclusion Taneli Riihonen Phase Noise in OFDM Repeaters 30 / 32
Conclusion Target: To understand the effect of spectral spreading on a full-duplex OFDM repeater link due to imperfect oscillator(s) Phase noise causes inter-carrier interference (ICI) Comparison of two different repeater designs 1. Using a single oscillator signal for down- and upconversion: Processing delay becomes a key factor for spectral spreading! 2. Separate oscillators for down- and upconversion Analysis and numerical results at three abstraction levels 1. Time-domain phase noise realizations vs. processing delay 2. Power spectral density of repeater s phase distortion process 3. Distribution of ICI power, and average transmission rate The transmit-side phase noise can partially revert the effect of receive-side distortion when processing delay is short enough Taneli Riihonen Phase Noise in OFDM Repeaters 31 / 32
Taneli Riihonen Phase Noise in OFDM Repeaters 32 / 32