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An Approximation to the Aliasing Effect, Part 2: Mitigation

In An Approximation to the Aliasing Effect, Part 1: The Origin I revisited some basic concepts about the aliasing effect, though I did it from a more visual than theoretical perspective. This second part deals with some mitigation techniques in use today. These techniques help improve signal-to-noise ratio (SNR) and effective number of bits (ENOB) of digitized signals.

Anti-aliasing filter
Let us continue with the same sample signals from the previous post and consider that all the information we are willing to extract ranges from DC-to-5 kHz (BW = 5kHz). As there are non-negligible frequency components above the bandwidth of interest, we must take care of them before sampling.

Bandwidth of interest and resulting aliased signal after sampling at 10 kHz.

Bandwidth of interest and resulting aliased signal after sampling at 10 kHz.

The first mitigation technique gets obvious here: Limit the input bandwidth. This is accomplished by using an analog filter before sampling.

There are several types of analog filters, and each filter approximation has its specific phase and ripple characteristics. One useful resource about filters is Chapter 8 of ADI’s Linear Circuit Design Handbook.

Selecting the most appropriated anti-aliasing filter could be a non-trivial task. The figure below depicts the frequency response for two filter approximations, Butterworth and Chebyshev Type-I, for two different filter orders, 2 and 4. The cutoff frequency has been selected at 5 kHz.

Butterworth and Chebyshev Type I filter examples.

Butterworth and Chebyshev Type I filter examples.

Note the pass-band ripple in the Chebyshev filter. I will not dig further into this subject as it was covered in Designing SAR ADC Anti-Aliasing Filters from Planet Analog’s Signal Chain Basics series.

The result of applying the second-order Butterworth filter with 5 kHz cutoff frequency on the sample signal is shown below. It can be seen how the 6 kHz tone has been reduced in amplitude though not completely removed. Note the 4 kHz tone has also been affected by the filter.

Plots showing the effect of applying the 2nd-order filter on the sample signal, shown with low frequency resolution for clarity.

Plots showing the effect of applying the 2nd-order filter on the sample signal, shown with low frequency resolution for clarity.

One obvious conclusion for this specific case is the need to use higher-order filters with sharper cutoff frequency edges.

Oversampling + digital filtering
Adding more analog filter stages can sometimes be impractical, even using readily available monolithic solutions. Raising the digitizer’s Nyquist frequency can be more appropriate and achievable in many cases. This technique is called Oversampling. Filtering the oversampled signal in the digital domain allows using higher-order filters. The next plot shows an example using a 25th-order Infinite Impulse Response (IIR) filter.

Plots showing the effect of applying the 25th-order filter on the sample signal, shown with low frequency resolution for clarity.

Plots showing the effect of applying the 25th-order filter on the sample signal, shown with low frequency resolution for clarity.

Note how the 6 kHz has been substantially attenuated while the 4 kHz tone alteration is almost negligible. The plot below shows in dB a segment from the same results set.

Segment taken from previous figure; the amplitude is represented in dB.

Segment taken from previous figure; the amplitude is represented in dB.

Oversampling has also been covered by our own Maithil Pachchigar in ADCs for High Dynamic Range: Successive-Approximation or Sigma-Delta?, Increase Dynamic Range With SAR ADCs Using Oversampling, Part 1, and Increase Dynamic Range with SAR ADCs Using Oversampling, Part 2.

Downsampling
There are applications where working with oversampled signals can compromise overall system performance. Downsampling can be used in those cases for reducing the sampling rate of the signal by an integer or rational factor.

It should be noted that downsampling a signal can be equivalent to the continuous time-sampling operation, especially in terms of the aliasing effect.

Decimation
The process of reducing the sampling rate, combining the operations of low-pass filtering, and downsampling is usually referred to as Decimation.

The decimation operation can be done also by working in the frequency domain instead of the time domain. It is accomplished by calculating the FFT of a signal segment, manipulating its Fourier transform, and then calculating the inverse FFT, though it is not appropriate for most real-time signal processing applications.

Hardware requirements
Albeit these digital-domain techniques prove effective for improving signal quality, we still find applications where they are limited to mid- and high-end data acquisition equipment and devices. Up to just a few years ago, digital signal processing applications were dominated by Digital Signal Processors.

We should note too that many readily available general-purpose processors support instruction sets featuring several commonly used DSP operations. Most modern core designs have also been optimized for achieving single-cycle instruction execution performance.

9 comments on “An Approximation to the Aliasing Effect, Part 2: Mitigation

  1. netrick
    November 26, 2014

    Instead of poles-only filters ahead of your ADC, I suggest to introduce a Poles-&-Zeros

    structure like Tchebitcheff-TypeII, also called Inverse-Trchebitcheff. You have a maximally

    flat behaviour in your Pass-Band below Fsampling/2, even better than Butterworth and 

    you can select your xmission Zeros to be on the disturbing narrow band disturbing 

    noises in the aliasing band between Fs/2 and Fs. I implement this in hybrid Analog-Digital

    filters built with Multiplying CMOS-DAC's. A good first trick is to position the first Zero

    right on Fs/2. The filters are built with cascaded 2nd order cells.

    An other solution is of course to drop the conventional approach altogether and go to

    an Hilbert-ADC with a dual synchronous sampling ADCa like AD 9703. The AA-filter is

    replaced by a “Quazor” of programmable Hilbert 90° Phase-Difference-Network with

    bands 1024_32_1 / 400_20_1 / 64_8_1, according to the Bedrosian tables.

    A system for wide-Bandwidth vibration analysis can thus be built with band selection

    409,6 kHz down to 1 Hz. 

  2. Victor Lorenzo
    November 29, 2014

    @netrick, thanks for reading and commenting the post, and welcome to PlanetAnalog.

  3. Victor Lorenzo
    November 29, 2014

    @netrick, selecting the filter approximation for the anti-aliasing filter requires evaluating the application requirements.

    Sometimes we need a maximally flat response in amplitud, some other times we need a maximally flat phase response and in some other cases the target goal is maximizing the reject band attenuation.

    All-pass filters are often used for modifying the phase response while keeping unity gain.

    There are implementations where simplicity and low cost of the digital part is a key requirement. That forces designers to move the complex signal conditioning and processing out to the analog part. Some other applications allow us to implement a high precision, variable gain, analog front-end with simple filtering stages and realise part or all of the linearization and filtering in the digital domain. In this cases FPGAs with embedded soft-core processors are great for the job. DSP hardware accelerator IP cores can be directly integrated and interfaced as peripherals of the soft-core processor.

  4. chirshadblog
    November 29, 2014

    @What if one of those loops get disconnected ?

  5. dassa.an
    November 29, 2014

    @chris: I guess there is very less chance of happening such an incident. Anyway there should be preventive action taken already

  6. netrick
    November 29, 2014

    I started to design ADC front-ends with active filters in the 60's. I started to design ADC 

    digital post processing in the early 80's. I was then in the pioneer team of ADI, when we

    acquired DSP-Systems and killed TRW in a snap. I know very well how to share the

    signal processing either in the analog side ahead or on the digital side at the end.

    I made some math reseach also to develop further the Paynter architectures. I also

    laugh sometimes when I read that FIR filtering had no analog equivalent before. 

    I can show them an old design we made at Lear with a tapped Zobel-Bessel LC 

    delay line and multipliers consisting in simple P_N transistor pairs. It worked very

    well associated to sonar signals to detect soviet subs.

  7. netrick
    November 29, 2014

    You mentioned in your presentation: “Our obvious conclusion for this specific case

    is the need to suse higher-order filters with sharper cut-off frequency edges “

    Why not use a filter which has the sharpest cut-off at band-edge and then the 

    shortest transition range. It is the Papoulis-Legendre filter, sometimes called

    “O” for Optimal-filter. It does not cost more than a Butterworth or a Tchebitcheff

    Type-I. And it is transitional behaviour is excellent.

  8. Victor Lorenzo
    November 29, 2014

    @netrick, thanks for mentioning the Optimum “L” filter.

    For filter orders higher than 2 this filter provides the advantages you mentioned and falls between a Chebyshev filter (sharp edge at cut off frequency) and a Butterworth filter (monotonic amplitude response), but its amplitude response is not exactly flat in the pass band.

    I agree with you, it could be useful for some applications.

    Thanks again for your contribution.

  9. Davidled
    December 1, 2014

    If adaptive filtering is used, 6kHz tone might be removed and 4 kHz tone might be recovered. Engineer might consider what kind of filtering could be used either high order flter or adaptive filter or filter combining the previous two methods.

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