Sampling and Multirate Techniques ForIq

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TAMPERE UNIVERSITY OF TECHNOLOGY/Telecommunications IQ/ IQ/ IQ/ IQ/1 M. Renfors 01.10.02

Sampling and Multirate Techniques forComplex and Bandpass Signals

Markku Renfors

Telecommunications LaboratoryTampere University of Technology, Finland

[email protected]

Topics:

•Complex signals and systems

•Sampling of complex signals

• Frequency translations using mixing and multirateoperations

•Real and I/Q sampling of bandpass signals

•Nonidealities in sampling and A/D-conversion




Complex Signals and Systems

In telecommunications signal processing, it is common to

use the notion of complex signals.

Continuous- and discrete-time complex signals are denotehere as

x t t jx t x k x k jx k R I R I ( ) ( ) ( ) ( ) ( ) ( )= + = +

In practical implementations, complex signals are nothin

more than two separate real signals carrying the real animaginary parts.

A complex linear time-invariant system is represented by twreal impulse responses

h k h k jh k R I ( ) ( ) ( )= +

or the corresponding two real-coefficient transfer functions

H e H e jH e j R

j I

j ( ) ( ) ( )ω ω ω = +

In the general case, to implement a complex filter for complex signal, four separate real filters need to bimplemented

y k x k h k x k jx k h k jh k x k h k x k h k j x k h k x k h k

R I R I

R R I I R I I R

( ) ( ) ( ) ( ( ) ( )) ( ( ) ( ))( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ))= ∗ = + ∗ +

= ∗ − ∗ + ∗ + ∗




Important Special Cases of Complex Signals




Sampling Theorem

The sampling theorem says that a (real or complex) lowpas

signal limited to the frequency band [-W, W] can representecompletely by discrete-time samples if the sampling rat(1/T ) is at least 2W .

In case of a complex signal, each sample is, of course, complex number.

In general, discrete-time signals have periodic spectrawhere the continuous-time spectrum is repeated arounfrequencies ± ± ±1 2 3T T T , , ,K

In case of complex signals, it is not required that the originasignal is located symmetrically around 0 if no overlappinoccurs in the frequency domain.

Any part of the periodic signal can be considered as th

useful part. This allows many possibilities for multiratprocessing of bandpass signals.

In general, the key criterion is that no distructive aliasingeffect occur.

0 f s 2 f s - f s -2 s

f




Real vs. Complex Discrete-Time Signals

Real signal:

Here 2W real samples per second are sufficient to representhe signal.

Complex signal:

Here W complex samples per second are sufficient.

• The resulting rates of real-valued samples are the same.

• However, the quantization effects may be quite different(Recall from the standard treatment of SSB that Hilberttransformed signals may be difficult.)

0 f s- f s

f

0- f s

f f s=W




Frequency Translation

One key operation is the frequency translation of a signaspectrum from one center frequency to another.

Conversions between baseband and bandpasrepresentations (modulation and demodulation) are speciacases of this.

We consider two different ways to do the frequenctranslation: mixing and multirate operations, i.e., decimatioand interpolation.

In case of multirate operations, we assume for simplicity thathe following two sampling rates are used:

low sampling rate: 1 NT

high sampling rate: 1T




Mixing for Complex Discrete-Time Signals

)()( 2 k xek y kT f j LOπ =

This produces a pure frequency translation of the spectrumby LO f .

Important special cases are:

T f f s LO 2

12/ ==

in which case the multiplying sequence is +1, -1, +1, -1, .

This case can be applied to a real signal withouproducing a complex result. Converts a lowpass signa

to a highpass signal, and vice versa.

T f f s LO 4

14/ ==

in which case the multiplaying sequence is+1, j, -1, -j, +1, j, ...

e j ω LO

kT

c

c+f LO

Special case: Real input

I

I

cos(ω LOkT

sin(ω LOkT )

I

Q




Complex Bandpass Filters

Certain types of complex filters based on Hilbe

transformers can be design using standard filter desigpackages, like Parks-McClellan routine for FIR filters.

Another way to get complex bandpass filters is througfrequency translations:

Real

prototypefilter:

Complexbandpassfilter:

Transformation for frequency response and transfer function

( ) ( ) T f j j j cc ze H z H e H e H π 2−−

→→

Transformation for block diagram:

→

T f j ceπ 2

If 1/T is an integer multiple of f c, this might be much easiethan in the general case, see the special cases of theprevious page.

0

f

0

f

f c




Example of a Complex Bandpass Filters:Frequency Translated FIR

Frequency translation by f s/4 => Analytic bandpass filter withpassband around f s/4.

Impulse response translated as:

h0, h1, h2, h3, h4, …, h N

⇓

h0, jh

1, -h

2, -jh

3, h

4, …, (j) N h

N

f s /4 f s /2

f

0

f s /4 f s /2

f

0




FIR Filter with Frequency Translation by f s/4

(i) Real input signal

If the filter length is odd and if it is a linear-phase design, thecoefficient symmetry can be exploited.

(ii) Complex input signal

• The possible coefficient symmetry can always beexploited.




Interpolation for Complex Signal

Sampling rate increase produces a periodic spectrum, andthe desired part of the spectrum is then separated by an(analytic) bandpass filter.




Decimation for Complex Signal

Sampling rate decrease produces aliasing, such that thoriginal band is at one of the image bands of the resultinfinal band.

The signal has to be band-limited to a bandwidth of NT/1before this operation can be done.




Combined Multirate Operations forComplex Signal

Combining decimation and interpolation, a frequency shift bn NT / can be realized, where n is an arbitrary integer.

It can be seen that the low sampling rate, limited to be highethan the signal bandwidth, determines the resolution of the

frequency translations based on multirate operations.

If, for example, a bandpass signal is desired to be translateto the baseband form, this can be done using multiartoperations if and only if the carrier frequency is a multiple othe low sampling rate.




Combining Mixing and Multirate Operations foComplex Signals

A general frequency shift of f n NT f O = + ∆ can be done in th

following two ways:

(1) Direct frequency conversion by O using mixing.

(2) Conversion using multirate operations by n NT followed by a mixing with ∆ (or vice versa).

The differences in these two approaches are due to thepossible filtering operations associated with the multiratoperations, and aliasing/reconstruction filters in case o

mixed continuous-time/discrete-time processing.

Assuming ideal filtering, these two ways would b

equivalent.




Example of Combining Mixing and MultirateOperations

Conversion from bandpass to baseband representation andecimation to symbol rate, i.e., I/Q-demodulation.

Assume that

- N =6, f 0=4/(6T )+ f ∆.- The required complex bandpass filter is obtained from

an FIR filter of length 50 by frequency translation.The following three ways are equivalent but lead to differencomputational requirements (the required real multiplicatiorates at input rate are shown; not exploiting possiblecoefficient symmetry):




Example of Combining Mixing and MultirateOperations (continued)

Notes:

(i) Complex bandpass filter, real inputs=> 100 real multipliers needed

(ii) Real lowpass filter, complex input to filter=>100 real multipliers

- Decimation can be combined efficiently with the filter

(iii) As (i) but decimation can be included efficiently with

the filter.- Mixing and LO generation done at lower rate andthus easier to implement.

Here we have not taken use of the possible coefficientsymmetry, which may reduce the multiplacation rates by 1/2

in all cases.




Frequency Translation for Real Signals

Mixing and multirate operations can be done in similar way for reasignals. The difference is that the two parts of the spectrum, on thpositive and negative frequency axis, and their images, must baccommodated in the spectrum.

(1) Mixing

Mixing produces two translated spectral components. The imag

band appearing on top of the desired band after mixing must bsuppressed before mixing.

(2) Multirate operations

In case of decimation, to avoid destructive aliasing effects, thsignal to be translated must be within one of the intervals

NT n

NT NT n

NT NT n

NT n ,

21or

21, −+

Otherwise distructive aliasing will occur. In the latter case, thspectrum will be inverted.

cos(ω LOkT)

-f c

c-f LO

0 c

0 c+f LO -f c+f LO-f c-f LO




Interpolation for Real Bandpass Signal




Decimation for Real Bandpass Signal




Example of Down-Conversion:I/Q-Demodulation

It is usually a good idea to keep the signal as a real signal along as possible, because after converting to complex formall subsequent signal processing operations require doublcomputational capacity compared to the corresponding reaalgorithms.




Real Bandpass Sampling

Down-conversion can also be implemented by sampling

bandpass signal. Any part of the periodic spectrum can beselected for further processing.

Concerning the sampling frequency, it is sufficient that no

aliasing appears on top of the desired band.

In general, the feasible sampling frequencies are determinefrom W , B (useful signal bandwidth), and f s. Minimumsampling frequency is B+W , which is adequate in the caswhere the center frequency of the desired signal is f s/4+kf s:

kf skf s - f s /2 kf s+f s /2

f

f

f c f c-W/2

BP

filter T/H

f s

kf skf s - f s /4 kf s+f s /4

f

W+B = f s




Quadrature Sampling

In this case we are sampling the complex analytic signaobtained by a phase-splitter:

The gain and phase imbalance analysis of quadrature down

conversion applies also to this case.

kf s (k+1)f s

f

BP

filter

90o T/H

T/H I

Q

analytic

bandpass signal

f s

f c f c-W/2




Second-Order Sampling

Quadrature sampling can be approximated by the followinstructure:

At the carrier frequency, the sampling time offsecorresponds exactly to the 90o phase shift. Farther awa

from the center frequency this is only approximative, but forelatively narrowband signals, it works. The nonideality cabe evaluated using the phase imbalance analysis.

BP

filter

T/H

T/H I

Q

f s

τ = 1/4 f c




Analysis of Second-Order Sampling

This system works perfectly at the carrier frequency buonly approximatively at other frequences. At frequency

∆+ f c , a time-shift of c f 41 corresponds to a phase

shift of

rads2

12

)(4

1

41

π π

+=⋅

+

∆

∆

cc

c

f

f

f f

f

We are actually dealing with phase imbalance and theimage rejection formula for quadrature mixing can beutilized (see slide 83080RA/16). The resulting imagerejection is:

⋅+

⋅−

=+

−=

∆

∆

2cos1

2cos1

cos1

cos1

π

π

φ

φ

c

c

f

f

f

f

R

Example: f c=1 GHz

∆ c f f ∆ Phaseimbalance Image rejection

0.1 MHz 0.0001 0.009o

82.1dB1 MHz 0.001 0.09

o 62.1 dB

10 MHz 0.01 0.9o 42.1 dB

100 MHz 0.1 9o 22.1 dB




Problems with Wideband Sampling

Analog to Digital Converter (ADC)Sampling a wideband signal, containing several channels isa tempting approach for designing a flexible radio receiver.However, there are some great challenges to do this.

The strongest signal in the ADC input signal band should bein the linear range of the ADC. When the desired signal is

weak, a large ADC dynamic range is needed, the resolutionof the converter has to be many bits, e.g., 14 ... 17 bits.

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 iunavoidable random variations in the sampling instantssampling aperture jitter. In bandpass sampling, threquirements for aperture jitter become very hard.




Quantization Noise in ADCs

In general, the maximum S/N-ratio for an A/D-converter iestimated by

nSNR s 2/log1076.102.6 10++= (dB)

where n is the number of bits B is the useful signal bandwidth

s is the sampling rate.

The last term takes into account the processing gain due toversampling in relation to the useful signal band. When thquantization noise outside that useful signal band is filteredaway, the overall qantization noise power is reduced by thefactor f s/2 B.

The number of additional bits needed to quantize awideband signal can be estimated by:

bits 6//log10 10 d B P P

where P B is the worst case power in the full band

P d is the minimum useful signal power.

Usually, in radio communications receivers, the worst casepower is determined from the adjacent channel or blockinsignal specifications




Spurious-Free Dynamic Range

Practical ADC's have also discrete spectral frequenccomponents, spurious signals (or spurs), in addition to thflat quantization noise.

In many applications, the spurious-free dynamic rangeSFDR, is the primary measure of the dynamic range of thconverter.




Track&Hold Circuit Nonidealities

Advanced bandpass sampling approaches could mean thawe are sampling a tens-of-MHz to GHz-range signal with relatively low sampling rate.

Noise Aliasing

Wideband noise at the sampling circuitry will be aliased tothe signal band. In case of bandpass sampling, aliasing

increases with increasing subsampling ( f c/ f s) factor.

Therefore, it is important to have a good noise figure for thtrack&hold circuit and/or to have sufficient amplification ithe analog front-end.




Aperture Jitter

Aperture jitter is the variation in time of the exact samplin

instant, that causes phase modulation and results in aadditional noise component in the sampled signal.

Aperture jitter is caused both by the sampling clock and thsampling circuit.




SNR Due to Sampling Jitter

The noise produced by aperture jitter is usually modeled awhite noise, which results in a signal-to-noise ratio of

=

aaj

T f SNR

max10

2

1log20

π

where f max is the maximum frequency in the sampler inpuand T

a is the rms value of the aperture jitter.

This model is derived for a sinusoidal input signal, buapplied also more generally, because no other models exist.

In critical test cases of the wideband sampling receiveapplication, the blocking signal is often defined as asinusoidal signal, and the model is expected to worreasonably well.

Example case:




About A/D-Conversion for SW Radio

It is obvious that the requirements for the T/H-circuitand A/D-converter are the main bottlenecks forimplementing receiver selectivity with DSP.

One promising A/D-converter technology in this contex

is the sigma-delta (Σ∆) principle.

- This principle involves low-resolution, high-speed

conversion in a noise-shaping configuration, togetherwith decimating noise filtering.

- In case of lowpass and bandpass sampling withsuitable fixed center frequency, this principle can becombined nicely with the selectivity filtering part of

the receiver.Noise filtering in basic ADC:

Noise filtering in sigma-delta converter:

f

f c

f

f c




Sigma-Delta Modulator

A Special quantization method – Different transfer functions for signal and noise

– Attenuates noise from the desired signal band, thus

in-band quantization noise is

Oversampling ratio has great effect

– Doubling the ratio Fs/2B, noise is decreased by

factor 3(2L+1) in dB.

– Number of quantization bits can be reduced

Noise can be filtered out by digital filters.

x[n]

e[n]

y[n]

- -

z -1

z -1

DAC

( ) ( ) ( ) ( ) z E z z X z z Y L L 1

1 −−−+=

( )( )

( )12222 2

1212

+

−

+

∆≅∫=

L

s

L B

BQe

F

B

Ldf f S

π σ

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Sampling and Multirate Techniques ForIq