Normalized Frequency
The Basics¶
What is Normalized Frequency
- Normalized Frequency \(f\) (unitless): (\(f=\frac{F}{F_s}\)\)
- Where
- \(F\) is the winding frequency of the Fourier Transform in Hz.
- \(F_s\) is the Sampling Frequency in Hz, and \(T_s\) is the Sampling Interval/Period in Seconds/Revolution.
- Typically restricted to a range of \(0\le f < 1\), because of the periodicity of the sinusoids, often restricted further due to Aliasing
- Sometimes rewritten equivalently as \(-\frac{1}{2}\le f < \frac{1}{2}\)
- Negative frequency causes \(-jsin(..)\) and \(cos(..)\). In other words, we rotate in the opposite direction as before.
- Where
- However, some usages define it in radians, aka Angular Frequency, by squashing \(j2\pi f\) into \(j \omega\): (\(\omega= 2\pi \frac{F}{F_s}=2\pi f\)\)
- Where \(0\le \omega < 2\pi\) radians
Where does Normalized Frequency come from?
- Normalized frequency is a result of the transition from a continuous-time to a discrete time representation.
- In CT, we might write a Complex Sinusoid as
\(\(\large e^{j2\pi Ft}\)\) - Where:
- \(F\) is the winding frequency in revolutions/second (Hz) (because of the addition of the \(2\pi\), the radian part is extracted out of F). For more info on this "winding frequency" see Fourier Transform.
- \(t\) is CT. Our input variable. \(Ft2\pi\) yields the current rotational position in radians.
- In DT, we want to abstract away \(t\) for \(n\), so we use the relation, \(\large t=T_{s}n\). This comes from the fact that each value of t is separated by an integer multiple (\(n\)) of sampling periods (\(T_{s}\))
- Typically, though, it's unwieldy to have to consider the sampling period of our system or our signal whenever we want to do this math
- \(\large t=\frac{n}{F_{s}}\)
- Substitute: \(\large \frac{F}{F_s}n\)
- Redefine terms: \(\large \frac{F}{F_{s}}=f\)
- In CT, we might write a Complex Sinusoid as
Bad times have a scientific value. These are occasions a good learner would not miss.
— Ralph Waldo Emerson