: Receiver Architectures
Outline Complex baseband equivalent of a bandpass signal Double-conversion single-quadrature (Superheterodyne) Direct-conversion (Single-conversion single-quad, homodyne, zero-) Weaver; Double-conversion double-quad Low- References
Complex Baseband Any passband waveform can be written in the following form: s p (t) =a(t) cos [ω c t θ(t)] s p (t) =a(t) cos ω c t cos θ(t) a(t) sin ω c t sin θ(t) s p (t) = s c (t) cos ω c t s s (t) sin ω c t s c (t) a(t) cos θ(t) =I(t) a(t) = s(t) = s c(t)s s(t) s s (t) a(t) sin θ(t) =Q(t) θ(t) = tan 1 s s(t) s c (t) We define the complex baseband signal and show that all operations at passband have a simple equivalent at complex baseband: s(t) =s c (t)js s (t) =I(t)jQ(t) { } s p (t) = Re s(t)e jω c t s = s p
Orthogonality I/Q An important relationship is the orthogonality between the modulated I and Q signals. This can be proved as follows (Parseval s Relation): x c (t) = s c (t) cos ω c t x s (t) = s s (t) sin ω c t <x c,x s >=< X c,x s >=0 <X c,x s >= X c (f)x s (f)df x c (t) = 1 (s c (t)e jω ct s c (t)e jω ct ) X c (f) = 1 (S c (f f c )S c (f f c )) cos θ = 1 (ejθ e jθ ) X s (f) = 1 j (S s (f f c ) S s (f f c )) <X c,x s >= 1 j ((S c (f f c )S c (f f c )) (S s (f f c ) S s (f f c ))) df
Orthogonality (Freq. Dom.) In the above integral, if the carrier frequency is larger than the signal bandwidth, then the frequency shifted signals do not overlap S c (f f c )S s (f f c ) 0 <X c,x s >= 1 j [ <X c,x s >= 1 j S c (f f c )S s (f f c ) 0 S c (f f c )S s (f f c )df [ S c (f)s s (f)df ] S c (f f c )Ss (f f c )df ] S c (f)ss (f)df =0 Due to this orthogonality, we can double the bandwidth of our signal my modulating the I and Q independently. Also, we have <u p,v p >=< u c,v c > <u s,v s >= Re (< u, v >)
Complex Baseband Spectrum Since the passband signal is real, it has a conjugate symmetric spectrum about the origin. Let s define the positive portion as follows: S p (f) =S p (f)u(f) Then the spectrum of the passband and baseband complex signal are related by: S(f) = S p (f f c ) S p (f) = S(f f c)s ( f f c ) Since: v(t) = s(t)e jω ct V (f) = S(f f c ) S p (t) = Re(v(t)) = v(t)v(t) S p (f) = V (f)v ( f) = S(f f c)s ( f f c )
The Image Problem LNA LO m r (t) cos(ω LO ω )t cos(ω LO )t = 1 m r(t) (cos(ω LO ω )t cos(ω )t) m i (t) cos(ω LO ω )t cos(ω LO )t = 1 m i(t) (cos(ω LO ω )t cos(ω )t) After low-pass filtering the mixer output, the is given by output = 1 (m i(t)m r (t)) cos(ω )t
Image Problem (Freq Dom) Complex Modulation (Positive Frequency) RF (ω ω 0 ) e jωlot RF (ω) RF (ω ω 0 ) IM (ω) IM (ω ω 0 ) e jωlot RF (ω) IM (ω ω 0 ) LO IM (ω) LO Complex Modulation (Positive Frequency) RF (ω) RF (ω ω 0 ) RF (ω) e jωlot IM (ω ω 0 ) e jωlot RF (ω ω 0 ) LO IM (ω) IM (ω) LO IM (ω ω 0 ) Real Modulation LO LO Complex modulation shifts in only one direction real modulation shifts up and down
Superheterodyne Architecture Off chip Image Filter Filter LNA Passive BPF The choice of the frequency dictated by: If the is set too low, then we require a very high-q image reject filter, which introduces more loss and therefore higher noise figure in the receiver (not to mention cost). If the is set too high, then subsequent stages consume more power (VGA and filters) Typical frequency is 100-00 MHz.
LO Planning in Superhet Off chip Image Filter Filter I LNA PLL Passive BPF LO 1 LO Q PLL 1 Two separate VCOs and synthesizers are used. The LO is fixed, while the RF LO is variable to down-convert the desired channel to the passband of the filter (SAW). This typically results in a 3-4 chip solution with many off-chip components. LO 1 should never be made close to an integer multiple of LO for any channel. The N th harmonic of the the fixed LO could leak into the RF mixer and cause unwanted mixing. LO LO 1 n LO nlo leaks into RF mixer
The ½ Problem LO ½ Assume that there is a blocker half-way between the LO and the desired channel. Due to second-order non-linearity in the RF front-end: [ m blocker (t) cos(ω LO 1 ω )t] =(m blocker (t)) (m blocker (t)) ) cos(ω LO ω )t If the LO has a second-order component, then this signal will fold right on top of the desired signal at : [ (mblocker (t)) ) cos(ω LO ω )t ] cos(ω LO )t =(m blocker (t)) cos(ω )t Note: Bandwidth expansion of blocker due to squaring operation.
Half- Continued DC DC ½ ½ If the stage has strong second-order non-linearity, then the half- problem occurs through this mechanism: [ m blocker (t) cos( 1 ω )t] =(m blocker (t)) (m blocker (t)) cos(ω )t This highlights the importance of frequency planning. One should select the by making sure that there is no strong half- blocker. If one exists, then the second-order nonlinearity must be carefully managed.
Dual-Conversion Single-Quad Off chip Image Filter Filter I LNA PLL Passive BPF LO 1 LO Q PLL 1 Disadvantages: Requires bulky off-chip SAW filters As before, two synthesizers are required Typically a three chip solution (RF,, and Synth) Advantages: Robust. The clear choice for extremely high sensitivity radios High dynamic range SAW filter reduces/relaxes burden on active circuits. This makes it much easier to design the active circuitry. By the same token, the power consumption is lower (< 5mA)
Complex Mixer A complex mixer is derived by simple substitution. Note that a complex exponential only introduces a frequency shift in one direction (no image rejection problems).
Hilbert Architecture A LNA C Image suppression by proper phase shifting. B RF = m r (t) cos(ω LO ω )t m i (t) cos(ω LO ω )t A = RF cos(ω LO t)= 1 m r(t) (cos(ω LO ω )t cos(ω )t) 1 m i(t) (cos(ω LO ω )t cos(ω )t) B = RF sin(ω LO t)= 1 m r(t) (sin(ω LO ω )t sin(ω )t) 1 m i(t) (sin(ω LO ω )t sin(ω )t) C = 1 m r(t)( cos(ω LO ω )t cos(ω )t) 1 m i(t)( cos(ω LO ω )t cos(ω )t) = A C = m r (t) cos(ω t) = A C = m i (t) cos(ω t)
Sine/Cosine Together Cosine Modulation LO LO 1/j Sine Modulation 1/j 1/j 1/j LO LO Delayed Sine Modulation LO LO Since the sine treats positive/negative frequencies differently (above/below LO), we can exploit this behavior A 90 phase shift is needed to eliminate the image 90 phase shift equivalent to multiply by j sign(f)
Hilbert Implementation Advantages: Remove the external image-reject SAW filter Better integration Requires extremely good matching of components (paths gain/phase). Without trimming/calibration, only ~40dB image rejection is possible. Many applications require 60dB or more. Power hungry (more mixers and higher cap loading). Note: A real implementation uses 45/135 degree phase shifters for better matching/ tracking. LNA A B D C
Gain/Phase Imbalance ) A = RF (1 α) cos(ω LO t φ 1 )= m r(t)(1 α) (cos(ω LO t ω t φ ) cos(ω t φ ) ) 1 m i(t)(1 α) (cos(ω LO t ω t φ ) cos(ω t φ ) ) B = RF (1 α) cos(ω LO t φ 1 )= m r(t)(1 α) (sin(ω LO t ω t φ ) sin(ω t φ ) ) 1 m i(t)(1 α) (sin(ω LO t ω t φ ) sin(ω t φ ) ) 1 C = m r(t)(1 α) ( cos(ω LO t ω t φ ) cos(ω t φ ) ) 1 m i(t)(1 α) ( cos(ω LO t ω t φ ) cos(ω t φ ) = A C = m r(t) ((1 α) cos(ω t φ ) (1 α) cos(ω t φ ) ) m i (t) ((1 α) cos(ω t φ ) (1 α) cos(ω t φ ) ) = m r (t) [cos(ω t) cos( φ ) α sin(ω t) sin( φ ] ) m i (t) [α cos(ω t) cos( φ ) sin(ω t) sin( φ ] )
Image-Reject Ratio IR(dB) = 10 log ( cos φ α sin ) φ α cos φ sin φ 10 8 6 4 Level of image rejection depends on amplitude/phase mismatch 0 1 3 4 Typical op-chip values of 30-40 db achieved (< 5, < 0.6 db) 0
RF/ Phase Shift, Fixed LO A A LNA RF RF B This requires a 90 degree phase shift across the band. It s much easier to shift the phase of a single frequency (LO). Polyphase filters can be used to do this, but a broadband implementation requires many stages (high loss)
Weaver Architecture RF = m r (t) cos(ω LO1 ω 1 )t m i (t) cos(ω LO1 ω 1 )t 1 = LO 1 RF A C LNA RF LO 1 LO = LO 1 = LO LO 1 RF = RF (LO 1 LO ) Eliminates the need for a phase shift in the signal path. Easier to implement phase shift in the LO path. Can use a pair of quadrature VCOs. Requires 4X mixers! Sensitive to second image. A LP F = cos ω LO1 t RF = m r cos(ω 1 )t m i cos(ω 1 )t B LP F = sin ω LO1 t RF = m r sin(ω 1 )t m i sin(ω 1 )t C LP F = A cos ω LO t = m r 4 cos(ω )t m i 4 cos(ω )t D LP F = B sin ω LO t = m r 4 cos(ω )t m i 4 cos(ω )t B D = C D = m r cos ω t
Direct Conversion (Zero-) LO LNA DC RF Let ω RF = ω LO = ω 0 m r (t) cos(ω RF )t cos(ω LO )t = 1 m r(t) (1 cos(ω 0 )t) DC The most obvious choice of LO is the RF frequency, right? = LO RF = DC? Why not? Even though the signal is its own image, if a complex modulation is used, the complex envelope is asymmetric and thus there is a mangling of the signal
Direct Conversion (cont) I LNA Q Use orthogonal mixing to prevent signal folding and retain both I and Q for complex demodulation (e.g. QPSK or QAM) Since the image and the signal are the same, the imagereject requirements are relaxed (it s an SNR hit, so typically 0-5 db is adequate)
Problems with Zero- LO = p(t) cos(ω LO t φ(t)) I LO LO = p(t) p(t) cos(ω LO t φ(t)) DC Dynamic DC Offset LNA Q Self-mixing of the LO signal is a big concern. LO self-mixing degrades the SNR. The signal that reflects from the antenna and is gained up appears at the input of the mixer and mixes down to DC. If the reflected signal varies in time, say due to a changing VSWR on the antenna, then the DC offset is time-varying
DC Offset 60 db Gain I LNA Q 1mV Offset 1V Offset DC offsets that appear at the baseband experience a large gain. This signal can easily saturate out the receive chain. A large AC coupling capacitor or a programmable DCoffset cancellation loop is required. The HPF corner should be low (khz), which requires a large capacitor. Any transients require a large settling time as a result.
Sensitivity to nd Order Disto Assume two jammers have a frequency separation of Δf: s 1 = m 1 (t) cos(ω 1 t) s = m (t) cos(ω 1 t ωt) (s 1 s ) =(m 1 (t) cos ω 1 t) (m (t) cos ω t) m 1 (t)m (t) cos(ω 1 t) cos(ω 1 ω)t LPF{(s 1 s ) } = m 1 (t) m (t) m 1 (t)m (t) cos( ω)t The two produce distortion at DC. The modulation of the jammers gets doubled in bandwidth. If the jammers are close together, then their intermodulation can also fall into the band of the receiver. Even if it is out of band, it may be large enough to saturate the receiver.
Sensitivity to 1/f Noise Since the is at DC, any low frequency noise, such as 1/f noise, is particularly harmful. CMOS has much higher 1/f noise, which requires careful device sizing to ensure good operation. Many cellular systems are narrowband and the entire baseband may fall into the 1/f regime! Noise Density 100 n i n i 1/f Slope, 0dB/dec Example: GSM has a 00 khz bandwidth. Suppose that the flicker corner frequency is 100 khz. The inband noise degradation is thus: v ave = 1 00kHz 100kHz 1kHz a f df 00kHz 100kHz bdf v ave = 1kHz 10 khz 1 00kHz 100 khz 00 khz [ a ln 100k 1k a = 1kHz 100 vi b = vi ] 1 b(00k 100k) = 00kHz (11.5a 100kb) =6.5v i
Low Architecture I LNA Complex Image Reject Filter Q Instead of going to DC, go a low, low enough so that the circuitry and filters can be implemented on-chip, yet high enough to avoid problems around DC (flicker noise, offsets, etc). Typical is twice the signal bandwidth. The image is rejected through a complex filter.
Double-Conversion Double-Quad A C D I LNA B E Q F The dual-conversion double-quad architecture has the advantage of de-sensitizing the receiver gain and phase imbalance of the I and Q paths.
Analysis of Double/Double Assuming ideal quadrature and no gain errors: RF = m r (t) cos(ω LO1 ω LO ω )t m i (t) cos(ω LO1 ω LO ω )t A = LPF{RF cos(ω LO1 t)} = 1 { mr (t) cos(ω LO ω )t m i (t) cos(ω LO ω )t B = LPF{RF sin(ω LO1 t)} = 1 { mr (t) sin(ω LO ω )t m i (t) sin(ω LO ω )t C = LPF{A cos(ω LO t)} = 1 { mr (t) cos(ω )t m i (t) cos(ω )t { mr (t) sin(ω )t D = LPF{A sin(ω LO t)} = 1 m i (t) sin(ω )t { mr (t) sin(ω )t E = LPF{B cos(ω LO t)} = 1 F = LPF{B sin(ω LO t)} = 1 m i (t) sin(ω )t { mr (t) cos(ω )t m i (t) cos(ω )t I = C F =(m r (t)m i (t)) cos(ω )t Q = D E =(m r (t) m i (t)) sin(ω )t
Gain Error Analysis RF = m r (t) cos(ω LO1 ω LO ω )t m i (t) cos(ω LO1 ω LO ω )t ( A = LPF{RF 1 a ) 1 cos(ω LO1 t)} = 1 ( 1 a ){ 1 mr (t) cos(ω LO ω )t m i (t) cos(ω LO ω )t ( B = LPF{RF 1 a ) 1 sin(ω LO1 t)} = 1 ( 1 a ){ 1 mr (t) sin(ω LO ω )t m i (t) sin(ω LO ω )t ( C = LPF{A 1 a ) cos(ω LO t)} = 1 ( 1 a )( 1 1 a ){ mr (t) cos(ω )t m i (t) cos(ω )t ( D = LPF{A 1 a ) sin(ω LO t)} = 1 ( 1 a )( 1 1 a ){ mr (t) sin(ω )t m i (t) sin(ω )t ( E = LPF{B 1 a ) cos(ω LO t)} = 1 ( 1 a )( 1 1 a ){ mr (t) sin(ω )t m i (t) sin(ω )t ( F = LPF{B 1 a ) sin(ω LO t)} = 1 ( 1 a )( 1 1 a ){ mr (t) cos(ω )t m i (t) cos(ω )t I = C F = (1 a 1 a )(m r (t)m i (t)) cos(ω )t Q = D E = (1 a 1 a )(m r (t) m i (t)) sin(ω )t The gain mismatch is reduced since due to the product of two small numbers (amplitude errors).
Phase Error Analysis RF = m r (t) cos(ω LO1 ω LO ω )t m i (t) cos(ω LO1 ω LO ω )t A = LPF{RF cos(ω LO1 t φ 1 )} = 1 { mr (t) cos(ω LO ω φ 1 )t m i (t) cos(ω LO ω φ 1 )t { mr (t) sin(ω LO ω φ 1 )t B = LPF{RF sin(ω LO1 t φ 1 )} = 1 m i (t) sin(ω LO ω φ 1 )t C = LPF{A cos(ω LO t φ )} = 1 { mr (t) cos(ω φ 1 φ )t m i (t) cos(ω φ 1 φ )t { mr (t) sin(ω φ 1 φ )t D = LPF{A sin(ω LO t φ )} = 1 m i (t) sin(ω φ 1 φ )t { mr (t) sin(ω φ 1 φ )t E = LPF{B cos(ω LO t φ )} = 1 F = LPF{B sin(ω LO t φ )} = 1 m i (t) sin(ω φ 1 φ )t { mr (t) cos(ω φ 1 φ )t m i (t) cos(ω φ 1 φ )t Q = D E =(m r (t) m i (t)) cos(φ 1 φ ) sin(ω )t The phase error impacts the I/Q channels in the same way, and as long as the phase errors are small, it has a minimal impact on the gain of the I/Q channels.
Double-Quad Low- A B I LNA C Q D Essentially a complex mixer topology. Mix RF I/Q with LO I/Q to form baseband I/Q Improved image rejection due to desensitization to quadrature gain and phase error.
References Fundamental of Digital Communication, U. Madhow, Cambridge 008 O. Shana a, EECS 90C Course Notes, 005. A.A. Abidi, Radio frequency integrated circuits for portable communications, Proc. of CICC, pp. 151-158, May 1994. A.A. Abidi, Direct conversion radio transceivers for digital communications, Proc. of ISSCC, pp. 186-187, Feb. 1995. J. Crols and M. Steyaert, A single-chip 900MHz receiver front-end with high performance low- topology, IEEE JSSC, vol. 30, no. 1, pp. 1483-149, Dec. 1995. B. Razavi, RF Microelectronics, Prentice Halls, 1998. J. Crols, M. Steyaert, CMOS Wireless Transceiver Design, Kluwer Academic Publishers, 1997. Practical RF System Design, Willian F. Egdan, Wiley-IEEE Press, 003.