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An Approximation to the Aliasing Effect, Part 1: The Origin

Previous articles in Planet Analog make mention of the “aliasing effect.” Most EEs agree in the importance of the aliasing effect as a noise source and take for granted that anti-aliasing filters are a key part in the signal chain path. But, what is the aliasing effect? Where does it come from? Why does oversampling followed by digital filtering and decimation improve the signal-to-noise ratio (SNR) and the effective number of bits (ENOB)? For understanding the aliasing effect we need to revisit some few relatively simple digital signal processing concepts.

For converting a continuous time signal from the analog domain to its representation in the digital domain we take “samples” at regular intervals. The inverse of the time between two consecutive conversions is usually known as “sampling rate.” The process of sampling can be considered like taking a snapshot of the signal for later conversion into its digital, discrete, representation. As a result of sampling the digitized signal is only “valid,” or defined, at the exact time of sampling (see picture below). As the sample resolution in the digital domain is finite, due to the limited number of output bits, another noise source to consider is the Quantization Noise.

Signal sampling example: sampled sinusoid.

Signal sampling example: sampled sinusoid.

Sampling in the continuous time domain is equivalent to multiplying by an impulse train (Dirac comb) with each impulse (Dirac Delta function) located at a multiple of the sample time.

Taking into account from Fourier Transform (FT) and signal processing theory:

  • The FT of a periodic impulse train (equally spaced at ts =1/fs ) in the time domain is also an impulse train in the frequency domain (equally spaced at fs ).
  • The product of two signals in the time domain translates to the convolution of their corresponding transforms in the frequency domain (convolution theorem).
  • As a result, the operation of sampling a signal translates in the frequency domain to the displacement of its spectrum (convolution with displaced impulses) centered in multiples of the sampling frequency fs . See the picture below.

Effect of sampling: a) Input signal spectrum, b) Sampled signal spectrum.

Effect of sampling: a) Input signal spectrum, b) Sampled signal spectrum.

Note the effect of displacing the signal's spectrum to multiples of fs: for frequencies above fs/2 the spectrums overlap. That is the origin of the aliasing effect.

Note also the sampled signal's spectrum is periodic though only the first part is shown in the figure and the spectrums overlapping effect (summation) is not properly represented.

The picture below depicts a practical simulation example using SciLab. The input signal is composed by three tones (1 kHz, 4.5 kHz, and 6 kHz) with amplitudes of 5V, 1V, and 1V, respectively. The upper part shows two periods of the signal, the lower two show the result of calculating the Fast Fourier Transform (FFT) over the sampled signal at two different sampling rates.

Aliasing example: The 6 kHz tone is 'aliased' at 4 kHz when sampled at 10 kHz (third plot).

Aliasing example: The 6 kHz tone is “aliased” at 4 kHz when sampled at 10 kHz (third plot).

As a result of the aliasing effect, when sampling at frequencies (fs) below that of the highest frequency component, all frequencies above fs/2 (Nyquist frequency) will be “folded” with respect to fs/2. These “aliased” frequencies cannot be distinguished from the non-aliased frequencies.

Now, with these few things in mind it should be easy to visualize several basic signal processing and signal conditioning methods that will produce an effective mitigation of the aliasing effect. In part 2 of this blog I will cover some of these methods.

8 comments on “An Approximation to the Aliasing Effect, Part 1: The Origin

  1. Victor Lorenzo
    September 12, 2014

    One thing to note in the last figure, I have intentionally created the spectrum plot with low frequency resolution as it helps for better visualization of the aliasing effect. Due to that the plot contains lobes instead of spectral lines though the input signal is composed of three pure tones.

  2. Davidled
    September 12, 2014

    FFT is most popular frequency analyzer method and even any instrument supporting the frequency response uses this method. Also, there is other method which is called Prony.  Frequency estimation could be conducted by Prony. But Prony might be more complex than FFT. That could be one of reasons why industry still has used FFT.     

  3. Victor Lorenzo
    September 13, 2014

    @DaeJ, thanks for mentioning Prony's method.

    For some analyzis, especially for image processing and compression, the Wavelet transform is also very valuable.

  4. Netcrawl
    September 13, 2014

    @Daej I agree with you, FFT is a powerrul tool in analyzing and measuring signals from plug-in  data acquisition devices, where you can effectively acquire time-domain signals, measure the frequency content and even convert the results to real-wrold units and displays. To perform FFT-based measurement you must (first) understand the fundamental issues and computational involved.

  5. Davidled
    September 13, 2014

    Usually, a FFT implementation would require many memories, but FFT with Time-multiplexed might consume less memory on pipeline architecture. I think that this method could be used in general processor.

  6. Netcrawl
    September 13, 2014

    @Daej moden spectrum analyzers are now almost exclusively Hybrid Superheterodyne-FFT based giving a significant increase in sweep time, however in some cases there still processing time to calculate the FFT.

  7. Victor Lorenzo
    September 14, 2014

    @DaeJ, it is possible to optimize the FFT calculation also for memory usage. Some DSP vendors provide specific implementations in assembler language for their DSP which make use of internal functionalities in order to make in-place FFT calculations.

  8. Victor Lorenzo
    September 14, 2014

    @Netcrawl, it is very difficult to acquire signals with components over 20~60GHz without using frequency convertion techniques.

    There are synthesizable FFT implementations for high performance FPGAs and ASICs. This speeds up FFT calculations and throughput.

    Many modern digital signal processors (DSP) include special addressing modes, like Butterfly, for significantly improving their FFT calculations performance.

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