Many of the systems we work on here at Cardinal Peak employ analog-to-digital conversion in one form or another. It seems like almost invariably some analog signal needs to be digitized or “sampled.” Sampling is the process of periodically looking at an analog signal at some sample rate, and then converting each measured value into a digital representation. The resulting samples have a resolution of some number of bits per sample; different applications will require different resolutions. Of course, the advantage of a stream of digital samples is that they can be efficiently processed by a DSP or other processor.
When analyzing sample data, the Discrete Fourier Transform is a powerful and handy tool, and we employ it often in the work we do for our clients. A recurring theme is that of power: how can the DFT be used to measure a signal’s power? Many of our clients’ engineers are familiar with the DFT and Parseval’s theorem, but need some reminding on how to apply it to power measurements in a real system. So this post briefly reviews that issue.
First, because the DFT pair can be expressed in different ways, it’s important to state which definition I’m using. I prefer the following commonly-encountered formulation:
X_{k}=\displaystyle\sum_{i=0}^{N-1}x_{n}e^{\dfrac{-i2\pi kn}{N}}
x_{n}=\dfrac{1}{N}\displaystyle\sum_{k=0}^{N-1}X_{k}e^{\dfrac{i2\pi kn}{N}}
In these equations the xn are the time domain samples of the analog signal, acquired at some sampling rate FS, and the Xn are the DFT coefficients.
Parseval’s theorem relates the time domain samples to the frequency domain samples via the following relationship:
\displaystyle\sum_{n=0}^{N-1}|x_{n}|^{2}=\dfrac{1}{N}\displaystyle\sum_{k=0}^{N-1}|x_{k}|^{2}
If the signal being sampled is a voltage, we can turn the left side into an approximation of the integral of the voltage squared as follows:
\displaystyle\sum_{n=0}^{N-1}|x_{n}|^{2} \dfrac{1}{F_{s}}=\dfrac{1}{NF_{s}}\displaystyle\sum_{k=0}^{N-1}|X_{k}|^{2}
Here we have multiplied both sides by 1 / FS because the samples are spaced by this amount of time from each other. Since we are processing N samples total, the time required to acquire these N samples is N / FS. Dividing both sides by this total duration, we get the average of the integral of the voltage squared over the acquisition period of the N samples.
\dfrac{1}{N}\displaystyle\sum_{n=0}^{N-1}|x_{n}|^{2}=\dfrac{1}{N^{2}}\displaystyle\sum_{k=0}^{N-1}|X_{k}|^{2}
Finally, the voltage was measured across a resistance of some value R, so the average power of the signal is given approximately by
\dfrac{1}{RN}\displaystyle\sum_{n=0}^{N-1}|x_{n}|^{2}=\dfrac{1}{RN^{2}}\displaystyle\sum_{k=0}^{N-1}|X_{k}|^{2}
Furthermore, it is reasonable to speak of the power carried at frequency
\dfrac{kF_{s}}{N}
as being
P_{k}=\dfrac{|X_{k}|^{2}}{RN^{2}}
Coming soon: noise floor and the DFT.