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.
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.
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.
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.