Sampling and Multirate Techniques for Complex and Bandpass Signals

Transcription

1 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/1 M. Renors, TUT/DCE Sampling and Multirate Techniques or Complex and Bandpass Signals Markku Renors Department o Communications Engineering Tampere University o Technology, Finland markku.renors@tut.i

2 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/2 M. Renors, TUT/DCE Contents Complex signals and systems o Basic concepts and deinitions o Analytic signals and Hilbert transorm o Frequency translation by mixing o Complex bandpass ilters Sampling and multirate DSP with complex and bandpass signals o Sampling theory or complex signals o Multirate processing o real and complex bandpass signals o Combining mixing and multirate operations or requency translation Bandpass sampling principles and related practical issues o Real bandpass sampling o Quadrature sampling o Second-order sampling o Sampling jitter and quantization noise o Sigma-delta ADC Complex and bandpass polyphase structures o Polyphase structures or decimation and interpolation o Real and complex ilter banks

3 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/3 M. Renors, TUT/DCE Complex Signals and Systems In telecommunications signal processing, it is common to use the notion o complex signals. Continuous- and discrete-time complex signals are denoted here as x( t) = x ( t) + jx ( t) x( k) = x ( k) + jx ( k) R I R I In practical implementations, complex signals are nothing more than two separate real signals carrying the real and imaginary parts. A complex linear time-invariant system is represented by two real impulse responses h( k) = h ( k) + jh ( k) R I or the corresponding two real-coeicient transer unctions jω jω jω R I He ( ) = H ( e ) + jh( e ) In the general case, to implement a complex ilter or a complex signal, our separate real ilters need to be implemented: y( k) = x( k) h( k) = ( xr( k) + jxi( k)) ( hr( k) + jhi( k)) = x ( k) h ( k) x ( k) h ( k) + j( x ( k) h ( k) + x ( k) h ( k)) R R I I R I I R

4 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/4 M. Renors, TUT/DCE Transorming Complex to Real Real Part -Operation Taking the real part o the complex signal produces mirror images o all the spectral components: Taking real part Aliasing takes place, i any o the mirror images overlaps with any o the original spectral components: Taking real part

5 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/5 M. Renors, TUT/DCE Transorming Real to Complex Analytic Signal and Hilbert Transorm Hilbert transormer is generally deined as an allpass linear ilter which shits the phase o its input signal by 9 degrees. The (anticausal) impulse and requency responses can be ormulated as continuous-time discrete-time 1, n even hht () t = hht ( n) = πt 2 /( π n), n odd j, HHT ( ) = jω j, ω π H ( ) + j, < HT e = < + j, π ω < Hilbert ilters can be used to construct analytic signals with only positive (or negative) requency content: input spectrum I INPUT OUTPUT output spectrum HT Q This structure is also called as phase splitter.

6 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/6 M. Renors, TUT/DCE Analytic Signal at Baseband Real signal spectrum, the corresponding analytic signal spectrum, and the analytic signal spectrum translated to be centered at -requency: X( ) W FILTERING W X( ) + jx( ) FREQUENCY TRANSLATION W W/ 2 W/2 In any practical design, there is a non-zero, usually symmetric, transition band around -requency. One practical example is VSB (vestigial side-band) modulation, the baseband model o which is obtained by a Hilbert ilter with symmetric transition band.

7 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/7 M. Renors, TUT/DCE Analytic Bandpass Signals Real bandpass signal and the corresponding ideal analytic bandpass signal: FILTERING W Other side o the desired signal spectrum suppressed by a practical phase splitter with inite attenuation (e.g., or improving image attenuation in case o I/Q downconversion): c To realize such a system, it is suicient to design an allpass ilter which approximates 9 phase shit in the passband and stopband with suicient accuracy, depending on the stopband attenuation requirement.

8 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/8 M. Renors, TUT/DCE Frequency Translation One key operation in communications signal processing is the requency translation o a signal spectrum rom one center requency to another. Conversions between baseband and bandpass representations (modulation and demodulation) are special cases o this. We consider two dierent ways to do the requency translation: mixing and multirate operations, i.e., decimation and interpolation. In case o multirate operations, we assume or simplicity that the ollowing two sampling rates are used: low sampling rate: = 1 s N NT high sampling rate: = 1 s T

9 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/9 M. Renors, TUT/DCE Mixing or Complex Discrete-Time Signals j2 ( ) LOkT j ( ) LOk yk e π ω = xk = e xk ( ) Special case with real input signal: e j ω LO k I I cos( ω Lo k) c sin( ω Lo k) + c LO I Q This produces a pure requency translation o the spectrum by. LO Important special cases are: LO = s / 2 = 1 2T in which case the multiplying sequence is +1, -1, +1, -1,... This case can be applied to a real signal without producing a complex result. Converts a lowpass signal to a highpass signal, and vice versa. LO = s / 4 = 1 4T in which case the multiplying sequence is +1, j, -1, -j, +1, j,...

10 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/1 M. Renors, TUT/DCE Complex Bandpass Filters Certain types o complex ilters based on Hilbert transormers can be designed using standard ilter design packages, like Parks-McClellan routine or FIR ilters. Another way to get complex bandpass ilters is through requency translations: Real prototype ilter: Complex bandpass ilter: Transormation or requency response and transer unction: ( jω ) ( j( ω ω )) ( ) ( j T ) c 2π H e H z H ze c H e Generic transormation or block diagram: c T T e j2πt c I 1/T is an integer multiple o c, this might be much easier than in the general case, see the special cases o the previous page.

11 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/11 M. Renors, TUT/DCE Example o a Complex Bandpass Filters: Frequency Translated FIR Frequency translation by s /4 => Analytic bandpass ilter with passband around s /4. s /4 s /2 s /4 s /2 Impulse response translated as: h, h 1, h 2, h 3, h 4,, h N h, jh 1, -h 2, -jh 3, h 4,, (j) N h N

12 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/12 M. Renors, TUT/DCE FIR Filter with Frequency Translation by s /4 (i) Real input signal I h h 2 h 4 T T T T T... h 1 h 3 h 5 Q (ii) Complex input signal I T T T T T h h 1 h 2 h 3 h 4 h 5... I Q Q h h 1 h 2 h 3 h 4 h 5 T T T T T... There are possibilities to exploit the possible coeicient symmetry (o linear phase FIR) in both cases.

13 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/13 M. Renors, TUT/DCE Contents Complex signals and systems o Basic concepts and deinitions o Analytic signals and Hilbert transorm o Frequency translation by mixing o Complex bandpass ilters Sampling and multirate DSP with complex and bandpass signals o Sampling theory or complex signals o Multirate processing o real and complex bandpass signals o Combining mixing and multirate operations or requency translation Bandpass sampling principles and related practical issues o Real bandpass sampling o Quadrature sampling o Second-order sampling o Sampling jitter and quantization noise o Sigma-delta ADC Complex and bandpass polyphase structures o Polyphase structures or decimation and interpolation o Real and complex ilter banks

14 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/14 M. Renors, TUT/DCE Sampling Theorem The sampling theorem says that a (real or complex) lowpass signal limited to the requency band [-W, W] can represented completely by discrete-time samples i the sampling rate (1/T) is at least 2W. In case o a complex signal, each sample is, o course, a complex number. In general, discrete-time signals have periodic spectra, where the continuous-time spectrum is repeated around requencies ± 1 T, ± 2 T, ± 3 T, 2 s s s 2 s In case o complex signals, it is not required that the original signal is located symmetrically around. Any part o the periodic signal can be considered as the useul part. This allows many possibilities or multirate processing o bandpass signals. In general, the key criterion is that no destructive aliasing eect occur on top o the desired part o the spectrum.

15 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/15 M. Renors, TUT/DCE Real vs. Complex Discrete-Time Signals Real signal: s Here 2W real samples per second are suicient to represent the signal. W s Complex signal: s =W s Here W complex samples per second are suicient. The resulting rates o real-valued samples are the same. However, the quantization eects may be quite dierent. (Recall rom the standard treatment o SSB that Hilbert-transormed signals may be diicult.)

16 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/16 M. Renors, TUT/DCE Interpolation or Complex Signal Sampling rate increase produces a periodic spectrum, and the desired part o the spectrum is then separated by an (analytic) bandpass ilter. N a) 1/NT COMPLEX BP-FILTER RESPONSE nnt / 1/ T nnt / b) 1/NT nnt /

17 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/17 M. Renors, TUT/DCE Decimation or Complex Signal Sampling rate decrease produces aliasing, such that the original band is at one o the image bands o the resulting inal band. The signal has to be band-limited to a bandwidth o 1 / NT beore this operation can be done without severe aliasing eects. N nnt / 1/ T 1/NT nnt / COMPLEX BP-FILTER RESPONSE

18 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/18 M. Renors, TUT/DCE Combined Multirate Operations or Complex Signal Combining decimation and interpolation, a requency shit by n/ N T can be realized, where n is an arbitrary integer. N M n/n- 1 S S n/n 1 S n/n 2 S It can be seen that the low sampling rate, limited to be higher than the signal bandwidth, determines the resolution o the requency translations based on multirate operations. I, or example, a bandpass signal is desired to be translated to the baseband orm, this can be done using multirate operations i and only i the carrier requency is a multiple o the low sampling rate. Using also simple requency translations (with coeicients +1, -1, +j, -j), the resolution is 1/(4NT).

19 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/19 M. Renors, TUT/DCE Combining Mixing and Multirate Operations or Complex Signals A general requency shit o n O = NT + Δ can be done in the ollowing two ways: (1) Direct requency conversion by O using mixing. (2) Conversion using multirate operations by n NT ollowed by a mixing with Δ (or vice versa). The dierences in these two approaches are due to the possible iltering operations associated with the multirate operations, and aliasing/reconstruction ilters in case o mixed continuous-time/discrete-time processing. Assuming ideal iltering, these two ways would be equivalent.

20 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/2 M. Renors, TUT/DCE Example o Combining Mixing and Multirate Operations Conversion rom bandpass to baseband representation and decimation to symbol rate, i.e., I/Q-demodulation. Assume that - N=6, =4/(6T)+ Δ. - The required complex bandpass ilter is obtained rom an FIR ilter o length 5 by requency translation. The ollowing three ways are equivalent but lead to dierent computational requirements (the required real multiplication rates at input rate are shown, not exploiting possible coeicient symmetry): (i) BPF (ii) LPF N N e j k (iii) BPF N e j k e j k Δ Case (i) Case (ii) Case (iii) Filter 1 1/6 1/6 Mixer 4 2 4/6 Total

21 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/21 M. Renors, TUT/DCE Example o Combining Mixing and Multirate Operations (continued) Notes: (i) Complex bandpass ilter, real inputs => 1 real multipliers needed or ilter (ii) Real lowpass ilter, complex input to ilter =>1 real multipliers needed or ilter - Decimation can be combined eiciently with the ilter. Utilizing coeicient symmetry is easiest in this case. (iii) As (i) but decimation can be included eiciently with the ilter. - Mixing and LO generation done at lower rate and thus easier to implement. Here we have not taken use o the possible coeicient symmetry, which may reduce the multiplication rates by 1/2 in all cases. In general, mixing is a memoryless operation, so upsampling and down-sampling operations can be commuted with it in block diagram manipulations.

22 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/22 M. Renors, TUT/DCE Frequency Translation or Real Signals Mixing and multirate operations can be done in similar way or real signals. The dierence is that the two parts o the spectrum, on the positive and negative requency axis, and their images, must be accommodated in the spectrum. (1) Mixing Mixing produces two translated spectral components (note that cos( ωt) = ( e j ω t + e jωt ) / 2). The image band appearing on top o the desired band ater mixing must be suppressed beore mixing. c c c LO c+ LO c LO c+ LO cos( ω LO t) (2) Multirate operations In case o decimation, to avoid destructive aliasing eects, the signal to be translated must be within one o the intervals n, n + 1 or 1, NT NT 2NT n NT 2NT n NT Otherwise destructive aliasing will occur. In the latter case, the spectrum will be inverted.

23 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/23 M. Renors, TUT/DCE Interpolation or Real Bandpass Signal wm ( ) xn ( ) N ym ( ) s N s X( ) s s /2 s /2 s W () k=3 k=2 k=1 k= k= k=1 k=2 k=3 Y () k=2 k=2 k=2 Y () k=3 k=3 k=3

24 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/24 M. Renors, TUT/DCE Decimation or Real Bandpass Signal xbp( n) xn ( ) N ym ( ) s s /N k=3 k=2 k=1 k= k= k=1 k=2 k=3 XBP( ) 3 s/(2n) s / N N s / 3 s/(2n) Y () /N s XBP( ) N s / 2 N s / 3 s/(2n) 3/(2 s N) 2/ N s Y () s / N N s /

25 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/25 M. Renors, TUT/DCE Analytic Filtering and Taking Real Part as Multirate Operations The transormations between real and complex signal ormats can be seen as multirate operations: Taking the real part eectively reduces the rate o realvalued samples by two. It produces mirror images, in contrast to the periodic images produced by decimation by two. In both cases, the new spectral components may all on top o the existing spectral components. I this operation ollows, e.g., an FIR ilter, considerable computational simpliications can be made by combining the real part- operation with the ilter in a cleaver way. There is no sense to compute samples that are thrown away by the real part operation!!. Analytic iltering (in any orm baseband, bandpass, ilter bank) increases the rate o real-valued samples by two. Mirror images are removed orm the spectrum, in contrast to the periodic images that are removed in interpolation.

26 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/26 M. Renors, TUT/DCE Example o Down-Conversion: I/Q-Demodulation N 4 PHASE SPLITTER 2 MATCHED FILTER 2 I Q 2/T 1/T It is usually a good idea to keep the signal as a real signal as long as possible, because ater converting to complex orm, all subsequent signal processing operations require (at least) double computational capacity compared to the corresponding real algorithms.

27 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/27 M. Renors, TUT/DCE Contents Complex signals and systems o Basic concepts and deinitions o Analytic signals and Hilbert transorm o Frequency translation by mixing o Complex bandpass ilters Sampling and multirate DSP with complex and bandpass signals o Sampling theory or complex signals o Multirate processing o real and complex bandpass signals o Combining mixing and multirate operations or requency translation Bandpass sampling principles and related practical issues o Real bandpass sampling o Quadrature sampling o Second-order sampling o Sampling jitter and quantization noise o Sigma-delta ADC Complex and bandpass polyphase structures o Polyphase structures or decimation and interpolation o Real and complex ilter banks

28 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/28 M. Renors, TUT/DCE Real Bandpass Sampling Down-conversion can also be implemented by sampling a bandpass signal. Any part o the periodic spectrum can be selected or urther processing. s c W/2 c T/H k /2 s s ks k s+s/2 Concerning the sampling requency, it is suicient that no aliasing appears on top o the desired band. In general, the easible sampling requencies are determined rom W, B (useul signal bandwidth), and c. Minimum sampling requency is B+W, which is adequate in the case where the center requency o the desired signal is k s ± s /4: W+B= s k /4 s s k s k + s s/4

29 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/29 M. Renors, TUT/DCE Quadrature Sampling In this case we are sampling the complex analytic signal obtained by a phase-splitter: T/H I s 9 T/H Q c W/2 c ANALYTIC BANDPASS SIGNAL k s ( k+ 1) s O course, in practise the image bands in between the shown periodic replicas are not completely attenuated but only suppressed to a level that is determined by the amplitude and phase imbalances in the phase splitter & sampler & ADC blocks. The gain and phase imbalance analysis o quadrature down-conversion applies also to this case.

30 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/3 M. Renors, TUT/DCE Second-Order Sampling Quadrature sampling can be approximated by the ollowing structure: T/H I s τ= 1/4 c T/H Q At the carrier requency, the sampling time oset corresponds exactly to the 9 o phase shit. Farther away rom the center requency this is only approximative, but or relatively narrowband signals, it works. The nonideality can be evaluated using the phase imbalance analysis.

31 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/31 M. Renors, TUT/DCE Analysis o Second-Order Sampling The second-order sampling concept works perectly at the carrier requency (ignoring the other sources o I/Q imbalance) but only approximately at other requencies. At requency c + Δ, a time-shit o 1 (4 c ) corresponds to a phase shit o 1 4 c π 2π 1 Δ = ra 1 + ds 2 ( ) c c + Δ We are actually dealing with phase imbalance and the image rejection ormula or quadrature mixing can be utilized. The resulting image rejection is: Δ π 1 cos 1 cosφ = 2 R = c 1+ cosφ 1+ cos Δ π c 2 Example: c =1 GHz Δ Δ Phase imbalance Image rejection c.1 MHz.1 ±.9 o 82.1dB 1 MHz.1 ±.9 o 62.1 db 1 MHz.1 ±.9 o 42.1 db 1 MHz.1 ±9 o 22.1 db

32 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/32 M. Renors, TUT/DCE Problems with Wideband Sampling Analog to Digital Converter (ADC) Sampling a wideband signal, containing several channels is a tempting approach or designing a lexible radio receiver. However, there are some great challenges to do this. The strongest signal in the ADC input signal band should be in the linear range o the ADC. When the desired signal is weak, a large ADC dynamic range is needed, the resolution o the converter has to be many bits, e.g., bits. MAGNITUDE REQUIRED DYNAMIC RANGE STRONG NEIGHBORING CHANNELS WEAK DESIRED CHANNEL Sampling The sampling to get a discrete time signal is done usually with a track-and-hold circuit (T/H). In practical sampling clocks and sampling circuits, there are unavoidable random variations in the sampling instants, sampling aperture jitter. In bandpass sampling, the requirements or aperture jitter become very hard.

33 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/33 M. Renors, TUT/DCE Quantization Noise in ADCs - 1 In general, the maximum S/N-ratio or an A/D-converter is SNR = 6.2n CFdB + 1log 1 [ s / kb] (db) where n is the number o bits CF db is the Crest Factor in db B is the useul signal bandwidth s is the sampling rate k 2 or baseband, 1 or bandpass sampling. The crest actor or a voltage signal is deined as the ratio o the peak absolute value and the RMS-value. The maximum SNR is achieved when the signal utilizes the A/D-converter s ull voltage range, which is assumed to be symmetric around DC. For a sinusoidal signal, the crest actor is 3 db, and or a bandpass signal, the crest actor is 3 db higher than that o the equivalent baseband signal. The last term takes into account the processing gain due to oversampling in relation to the useul signal band. When the quantization noise outside that useul signal band is iltered away, the overall quantization noise power is reduced by the actor s /kb. Considering the choice between (real) baseband and bandpass sampling, the dierence is due to the 3 db higher crest actor in the bandpass sampling case. The k parameter is just due to the deinition o bandwidth. The bandwidths o B in baseband model 2B in bandpass model contain the same amount o signal power and quantization noise power.

34 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/34 M. Renors, TUT/DCE Quantization Noise in ADCs - 2 Let s consider now the case o complex I/Q sampling. In most cases, hal o the overall signal power is captured by both o the ADCs in the I and Q branches. Thus the signal to quantization noise power ratio is the same in both ADC s as that o the overall complex signal. Also the processing gain is the same as in the real case, i the bandwidths are deined in the same way. However, the ull peak value o the waveorm may appear in both o them. Thus the crest actor should be deined as the ratio o the peak complex signal magnitude divided by the rms value o the I or Q signal. The number o additional bits needed to quantize a wideband signal, containing strong spectral components in addition to the wanted signal, can be estimated by: CF 2 1log B P B 1 / 6 bits CF 2 d Pd where P B is the worst case power in the ull band P d is the minimum useul signal power CF B is the crest actor in the worst case test situation B CF d is the crest actor o the desired signal. Usually, in communications receivers, the worst case power is determined rom the adjacent channel or blocking speciications. For many types o communications signals (e.g., CDMA, OFDM), the crest actor is much higher than that o the sinusoid, which might be a valid assumption in the blocking test situation.

35 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/35 M. Renors, TUT/DCE Spurious-Free Dynamic Range Practical ADC's have also discrete spectral requency components, spurious signals (or spurs), in addition to the lat quantization noise. In many applications, the spurious-ree dynamic range, SFDR, is the primary measure o the dynamic range o the converter.

36 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/36 M. Renors, TUT/DCE Track&Hold Circuit Nonidealities Advanced bandpass sampling approaches could mean that we are sampling a tens-o-mhz to GHzrange signal with a relatively low sampling rate. Noise Aliasing Wideband noise at the sampling circuitry will be aliased to the signal band. In case o bandpass sampling, aliasing increases with increasing subsampling ( c / s ) actor. Basically, the noise igure depends on the subsampling actor. Thereore, it is important to have a good noise igure or the track&hold circuit and/or to have suicient ampliication in the analog ront-end.

37 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/37 M. Renors, TUT/DCE Aperture Jitter Aperture jitter is the variation in time o the exact sampling instant, that causes phase modulation and results in an additional noise component in the sampled signal. Aperture jitter is caused both by the sampling clock and the sampling circuit.

38 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/38 M. Renors, TUT/DCE SNR Due to Sampling Jitter The noise produced by aperture jitter is usually modelled as white noise, which results in a signal-to-noise ratio o SNR aj 1 = 2log 1 2 π max T where max is the maximum requency in the sampler input and T a is the RMS value o the aperture jitter. This model is derived or a sinusoidal input signal, but applied also more generally. In critical test cases o the wideband sampling receiver application, the blocking signal is oten deined as a sinusoidal signal, and the model is expected to work reasonably well. The processing gain due to oversampling eects in the same way as in case o quantization noise. Example o Sampling Jitter Eects - 14 bits - 1 ps RMS jitter a Aperture jitter eect Quantization eect Joint eect SNR in db Frequency in MHz

39 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/39 M. Renors, TUT/DCE About A/D-Conversion or SW Radio It is obvious that the requirements or the T/H-circuit and A/D-converter are the main bottlenecks or implementing receiver selectivity with DSP. One promising A/D-converter technology in this context is the sigma-delta (ΣΔ) principle. - This principle involves low-resolution, high-speed conversion in a noise-shaping coniguration, together with decimating noise iltering. - In case o lowpass and bandpass sampling with suitable ixed center requency, this principle can be combined nicely with the selectivity iltering part o the receiver. Noise iltering in basic ADC: c Noise iltering in sigma-delta converter: c

40 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/4 M. Renors, TUT/DCE Sigma-Delta Modulator - 1 en [ ] xn [] u[n] yn [ ] DAC This is a second-order sigma-delta modulator. The sequence e[n] models the quantization noise o a single-bit (or ew-bit) ADC included in the loop. In an Lth-order modulator, the irst integrator is repeated L-1 times. The system has dierent transer unctions or input signal and quantization noise. They can be analyzed as ollows: 1 L 1 1 z 1 z U( z) = 1 1 ( X( z) Y( z) ) Y( z) 1 1 z 1 z 1 z Y( z) = U( z) + E( z) Now Y(z) can be solved as ( ) L 1 1 = ( ) + ( 1 ) ( ) Y z z X z z E z The signal and noise transer unctions can be identiied rom this expression as: 1 STF = z NTF 1 (1 ) L = z

41 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/41 M. Renors, TUT/DCE Sigma-Delta Modulator - 2 NTF attenuates noise rom the desired signal band. The inband quantization noise variance becomes σ 2 e B 2 SQ( ) NTF( ) d B 2 B 2 B Δ 2L Δ 2L sin ( π / s) d ( π / s) d 12 s 12 B s B 2L L = = Δ π 2B = 12 ( 2L+ 1) s Oversampling ratio has great eect: Doubling the oversampling ratio F s /2B, the noise is decreased by the actor 3(2L+1) in db. The number o quantization bits can be reduced by increasing the oversampling ratio. Noise can be iltered out by decimating digital ilters.

42 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/42 M. Renors, TUT/DCE Contents Complex signals and systems o Basic concepts and deinitions o Analytic signals and Hilbert transorm o Frequency translation by mixing o Complex bandpass ilters Sampling and multirate DSP with complex and bandpass signals o Sampling theory or complex signals o Multirate processing o real and complex bandpass signals o Combining mixing and multirate operations or requency translation Bandpass sampling principles and related practical issues o Real bandpass sampling o Quadrature sampling o Second-order sampling o Sampling jitter and quantization noise o Sigma-delta ADC Complex and bandpass polyphase structures o Polyphase structures or decimation and interpolation o Real and complex ilter banks

43 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/43 M. Renors, TUT/DCE Polyphase Decomposition Any FIR ilter transer unction can be expressed as L 1 H( z) = z ihi ( zl) i= In this case, the polyphase branch ilters are FIR ilters o length M / L or M / L, where M is the length o the original FIR ilter. Example: Polyphase structure with L=2: H (z 2 ) = z 1 H 1 (z 2 ) + Also certain types o IIR ilters can be expressed in the above orm. Especially, the Nth-band IIR ilters have polyphase decomposition where the polyphase branches are allpass ilters.

44 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/44 M. Renors, TUT/DCE Applications o Polyphase Structure 1 Complex Bandpass Filters Including a requency shit by k s /L, the polyphase ilter can be represented as L 1 H( z) = ( z e jk2 π / L) i Hi ( zl) i= So we can see that the complex multipliers or requency shiting do not appear in the polyphase branches, but only in the separate unit delay chain. I this part o the structure is on the output side, then just replicating this structure with dierent values o k, we obtain bandpass ilters with the same input but dierent center requencies, and sharing the main part o the computations. In this way we can implement ilter banks that split the input signal into a number o narrower signal bands. Polyphase structures are mostly considered in multirate signal processing applications, as discussed on the ollowing pages. Here we want to point out that they can be useul even in cases where sampling rate conversion is not involved, e.g., or implementing complex bandpass ilters and ilter banks.

45 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/45 M. Renors, TUT/DCE Applications o Polyphase Structure 2 Decimation and Interpolation Filters The polyphase structure with sampling rate reduction by two looks as ollows: H (z 2 ) 2 z 1 H 1 (z 2 ) Using the so-called noble identities, the structure can be developed as ollows: 2 H (z) z 1 2 H 1 (z) The unit delay and sampling rate reduction blocks just model the operation o connecting even-indexed input samples to the upper branch and odd-indexed ones two the lower branch.

46 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/46 M. Renors, TUT/DCE General Polyphase Structures Decimation by L: Interpolation by L:

47 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/47 M. Renors, TUT/DCE Bandpass Polyphase Structures and Filter Banks As mentioned already, we can easily embed in the polyphase structure the complex coeicients needed or requency translation. In the decimation case, the bandpass polyphase structure looks as ollows: with wik, = e jk(2 π ) i / L, i = 1, 2,..., L 1. A uniorm ilter bank can be obtained by repeating the complex multipliers and output adder with dierent values o k. I all (or even several) channels are need, this can be done very eiciently using FFT operation. Such a structure is generally called DFT ilter bank. The decimation case corresponds to analysis bank, in which the wideband signal is divided into L subchannels with sample rate s /L. Likewise, the interpolation case corresponds to a synthesis bank, in which several narrowband signals are combined into a single wideband signal.

48 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/48 M. Renors, TUT/DCE Real Critically-Sampled Filter Banks In critically sampled L-channel ilter banks, the sampling rate conversion actor is L. For every L input samples o a (decimating) analysis bank, one sample is generated to each o the low-rate subbands. An analysis-synthesis system includes analysis and synthesis banks, connected by a subband processing stage. L L L L x(n) L-channel Analysis Filter Bank Processing L-channel Synthesis Filter Bank L L The perect reconstruction property means that, in the absence o subband processing, the synthesis bank output is an exact delayed replica o the analysis bank input. With proper ilter bank design, perect reconstruction can be achieved even in the critically sampled case, even though aliasing takes place in the subband signals. These must have considerably overlapping transition bands in practical designs. Cosine-modulated ilter banks (CMFBs) are a common and eicient solution or real critically-sampled uniorm ilter banks.

49 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/49 M. Renors, TUT/DCE Filter Bank vs. FFT/IFFT Probably the computationally most eicient ilter banks are FFT (as analysis bank) and IFFT (as synthesis bank). The beneit o ilter banks is better requency selectivity o the subbands, at the cost o somewhat higher complexity. Example: Subband requency responses or L=16 channels (a) FFT: Amplitude in db Frequency ω / π (b) CMFB, prototype ilter length=8*l, 1% roll-o: Amplitude in db Frequency ω / π The uniorm, modulation-based ilter banks considered here consist o the prototype ilter in polyphase or lattice orm, and a transorm block, e.g., discrete cosine/sine transorm, or DFT/FFT.

50 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/5 M. Renors, TUT/DCE Complex Critically-Sampled Filter Banks - 1 The structure o the real critically sampled ilter bank cannot be generalized to the complex case as such. Using L-channel banks and sampling rate conversion by L, together with complex subband signals leads to perect reconstruction only in case o the DFT-IDFT pair. The solution is to use 2M-channel banks, sampling rate conversion by M, and real subband signals. Exponentially modulated ilter bank (EMFB): x(n) H (z) M Re[ ] M F (z) H 1 (z) M Re[ ] M F 1 (z) H 2M 1 (z) M Re[ ] M F 2M 1 (z) ˆx(n)

51 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/51 M. Renors, TUT/DCE Complex Critically-Sampled Filter Banks 2 In the EMFB case, the sub-channel impulse responses o the synthesis bank are obtained rom a low-pass prototype ilter h p (n) as ollows: ± c s 2 L π k ( n) = k ( n) ± j k ( n) = hp( n) exp ± j n+ k + L 2 2 L Here the +/- signs are or the subbands on the positive/negative sides o the spectrum, respectively. The analysis bank impulse responses are related to the synthesis bank as ollows: ± c s c k k k k ( ) = ( 1 ) ( 1 ) = ( ) ( ) h n N n j N n h n j h n where N is the length o the prototype ilter. Critically-sampled perect-reconstruction EMFB systems can be obtained by using the same prototype ilters as or CMFBs. s k

52 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/52 M. Renors, TUT/DCE Complex Critically-Sampled Filter Banks - 3 EMFBs can be implemented eiciently using CMFBs and SMFBs as building blocks (also an eicient FFT-based structure is available): x (m) Analysis Synthesis x I (n) CMFB x M 1 (m) CMFB ˆx I (n) x 2M 1 (m) x Q (n) Analysis Synthesis ˆx Q (n) SMFB x M (m) SMFB

53 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/53 M. Renors, TUT/DCE Complex Critically-Sampled Filter Banks 4 EMFB s use so-called odd channel stacking arrangement: F M 1 ( ω) F 1 ( ω ) + + F ( ω ) ( ω ) ( ω ) F F 1 + F M 1 ( ) ω π π Ater decimation, the subband signals are centered at π/2 or π/2, and the other side o the spectrum is empty. This is also important or perect reconstruction, since now there is room or the mirror image produced by the real-part operation. Then, only the transition bands o the two spectral components are overlapping. What basically happens is that a real subband signal is constructed rom its analytic version. Also even channel stacking arrangement can be used in complex ilter banks. In this case, the subbands are centered at kπ/m, and ater decimation they are centered at or π. In this case, the real part operation cannot be done as such or the subband signals, because the mirror image would all on top o the decimated subband signal. One well known class o complex ilter banks using even channel stacking are the modiied DFT (MDFT) ilter banks. MDFT banks use slightly more complicated processing o the subband signals to achieve perect reconstruction. This leads to more complicated relation between the original spectrum and the subband signal spectra.

54 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/54 M. Renors, TUT/DCE Transmultiplexers A transmultiplexer system includes also analysis and synthesis banks, but in the reverse order. They are mostly applied in communication systems or TDM FDM transmultiplexing and or multicarrier modulation, as an alternative to the widely adopted IFFT/FFT based OFDM system. In a transmultiplexer, a number o low-rate symbol sequences are modulated to subcarriers and combined into a wideband signal or transmission. In the analysis bank o the receiver, the subcarrier signals are again separated. In an ideal transmultiplexer, in case o an ideal noise-ree channel, the output subcarrier sequences are exactly the same as the input sequences. Perect-reconstruction CMFBs, EMFBs, and MDFT banks can directly be used to get ideal transmultiplexers. The main beneits o ilter bank based multicarrier modulation over OFDM are due to the high requency selectivity: the system is very robust to narrowband intererences and the overall spectrum has very sharp transition bands, allowing to use very narrow guard-bands in requency domain. Also, the receiver ilter bank can implement a considerable part o the channel selection iltering in a lexible manner.

55 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/55 M. Renors, TUT/DCE EMFBs as Transmultiplexer Real transmultiplexers have real baseband signal, whereas in the complex case, the wideband baseband signal is complex, and it can be directly modulated to the RF carrier in a spectrally eicient way. CMFBs and SMFBs can again be used as building blocks or critically-sampled complex transmultiplexers: X + ( z ) X + 1 ( z) + M M M M + X + ( z) X + 1 ( z) Synthesis: Cosine Modulated Filter Bank Analysis: Cosine Modulated Filter Bank I + X M 1 ( z) M M + X M 1 ( z) X ( z) X 1 ( z) M M Q Channel M M X ( z) X 1 ( z) Synthesis: Sine Modulated Filter Bank Analysis: Sine Modulated Filter Bank X M 1 ( z) M M X M 1 ( z) Here we have again 2M subchannels using sampling rate conversion actor M. In the critically-sampled case, the subchannel signals are real. In the modulation sense, VSB modulation or oset-qam modulation take place or the subcarrier signals.

56 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/56 M. Renors, TUT/DCE Complex Oversampled Transmultiplexer In a analysis-synthesis or transmultiplexer system, aliasing and imaging are taking place at dierent processing stages, but the eects cancel each other and perect reconstruction is achieved in ideal conditions. However, in case o transmultiplexers, non-ideal, requency-selective transmission channel destroys the orthogonality o the subcarrier signals, introducing intercarrier intererence (ICI, crosstalk) between adjacent subchannels and intersymbol intererence (ISI) within each subchannel. These eects cannot be compensated eectively by subband-wise processing. Thus, in the critically sampled case, the channel equalizers ater analysis bank have to include connections between adjacent subchannels. Usually in the ilter bank design, the roll-o is no more than 1 %, i.e., the overall subchannel bandwidth is twice the channel spacing, or less. I the subchannel signals can be obtained in 2x -oversampled orm, then they are essentially alias ree (the alias components are attenuated at least by the stopband attenuation o the channel ilters). Then, in principle, it becomes possible to do the channel equalization or each subcarrier independently o the others. In the EMFB, there is a simple way to obtain 2x oversampling or the subchannel signals: using the complex subchannel signals instead o the real ones that are suicient in the critically sampled case.

57 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/57 M. Renors, TUT/DCE EMFB-Based Transmultiplexer with 2x-Oversampled Analysis Bank and Per- Subcarrier Equalization + X k ( m) 1 2 CMFB Synthesis Re{. } CMFB Analysis SMFB Analysis I Q c,..., c + + k,, ck, 1 + k + L Re{. } + X k ( m) H lp (z) X k ( m) 1 2 SMFB Synthesis j Equivalent lowpass channel Im{. } CMFB Analysis SMFB Analysis Q I c,..., c k,, ck, 1 k + L Re{. } X k ( m) Here the synthesis bank is critically sampled, analysis bank is oversampled by 2, and ater subchannel equalizers, symbol-rate sequences are obtained by taking the real part o the equalizer output (usually, the real part operation can be eiciently combined with the equalizer).

58 Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/58 M. Renors, TUT/DCE Complex Oversampled Analysis-Synthesis System One good application o analysis-synthesis FB systems is in requency-domain channel equalization (FDE) in communication systems. FFT-IFFT -based FDEs have received a lot o attention recently because, in wideband transmission, they result in signiicantly lower complexity than time-domain equalization. FFT-based FDE s use commonly a cyclic preix, in the same way as OFDM, to achieve simple and robust system. Using analysis-synthesis FB system in FDE, instead o FFT-IFFT, has a number o beneits due to the good requency selectivity, as mentioned already in the multicarrier context. Also here, in the critically sampled case, subband-wise equalization results in poor perormance, and cross-connections between subbands would be needed. A 2x -oversampled EMFB-type analysis bank can be realized in the same manner as in the transmultiplexer case. This makes it easible to do the channel equalization subband-wise. Then the real parts o the subband equalizer outputs are connected to a critically sampled synthesis bank. Naturally, analysis-synthesis systems with oversampled subband processing could be a useul tool or various other applications.