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I am working on DSP and finding difficulty to understand the term Normalized frequency often used with DFT & DTFT.

What is normalized frequency in DSP ? and how it is different from analog frequency ?

What is significance to normalize frequency in DSP ?

Why limit of normalized frequency is 2π ?

How FFT deals with normalized frequency ?

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Normalized frequency is frequency in units of cycles/sample or radians/sample commonly used as the frequency axis for the representation of digital signals.

When the units are cycles/sample, the sampling rate is 1 (1 cycle per sample) and the unique digital signal in the first Nyquist zone resides from a sampling rate of -0.5 to +0.5 cycles per sample. This is the frequency equivalent of representing the time axis in units of samples instead of an actual time interval such as seconds.

When the units are radians/sample, the sampling rate is $2\pi$ ($2\pi$ radians per sample) and the unique digital signal in the first Nyquist zone resides from a sampling rate of $-\pi$ to $+\pi$.

How this comes about can be seen from the following expressions:

For an analog signal given as $$x(t)=\sin(2\pi F t)$$ where F is the analog frequency units in Hz,

When sampled at a sampling frequency of $F_s$ Hz, the sampling interval is $T_s=1/F_s$ so the signal after being sampled is given as:

$$x(nT_s)=\sin(2\pi F nT_s) = \sin\left(\frac{2\pi F}{F_s}n\right)$$

Where the units of normalized frequency, either $\frac{F}{F_s}$ in cycles/sample or $\frac{2\pi F}{F_s}$ in radians/sample is clearly shown.

This is illustrated below using $\Omega = 2\pi F$

Update: As @Fat32 points out in the comments, the units for sampling rate $F_s$ in the figure below should be "samples/sec" in order for the normalized frequency to become radians/sample.

Normalized Frequency

To visually see the concept of "radians/sample" (and most other DSP concepts dealing with frequency and time) it has helped me considerably to get away from viewing individual frequency tones as sines and/or cosines and instead view them as spinning phasors ($e^{j\omega t} = 1 \angle (\omega t)$) as depicted in the graphic below, which shows a complex phasor spinning at a rate of 2 Hz and it's associated cosine and sine (being the real and imaginary axis). Each point in a DFT is an individual frequency tone represented as a single rotating phasor in time. Such a tone in an analog system would continuously rotate (counter-clockwise if a positive frequency and clockwise if a negative frequency) at F rotations per second where F is the frequency in Hz, or cycles/second. Once sampled, the rotation will be at the same rate but will be in discrete samples where each sample is a constant angle in radians, and thus the frequency can be quantified as radians/sample representing the rate of rotation of the phasor.

Euler's Identitiy

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    $\begingroup$ I was only half way through my coffee! Complete mistake, fixed now $\endgroup$ – Dan Boschen Jun 7 '17 at 14:11
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    $\begingroup$ The FFT typically uses yet another unit for the frequency axis referred to as the Frequency Index which goes from 0 to N-1 where N is the number of samples used in the FFT . This maps to Normalized Frequency by equating N to 1 cycle/sample. Thus if you divide the FFT Frequency by N you get Normalized Frequency in cycles/sample. For example if I have 10 samples in the FFT in a system sampled at 100 Hz, the frequency bins in the FFT result will go from 0, 10, 20 ....90 Hz. N-1= 9 and 100 Hz represents 1 sample per cycle. Hope that helped. $\endgroup$ – Dan Boschen Jun 7 '17 at 15:21
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    $\begingroup$ Yes I see- the sampling rate should be given in units of "Samples per Second" to make that work, good point @Fat32! Consistent units are important. $\endgroup$ – Dan Boschen Jun 7 '17 at 15:45
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    $\begingroup$ @user6363 The sampling rate is 1 cycle/sample means when you use Normalized frequency then whatever the sampling rate is becomes 1 (cycle per sample), for example if the sampling rate is 100 MHz, then 100 MHz maps to 1, and a tone at 25 MHz for example would map to 0.25 (cycles/sample). When the units are radians/sample, the 100 MHz sampling rate would map to $2\pi$ and the tone at 25 MHz would map to $0.5\pi$. In my example the waveform extends in bandwidth from +6π/20 on a radians/sample normalized scale. If the sampling rate was 100 MHz, then this would be +/-3/20 *100 MHz = +/-15MHz $\endgroup$ – Dan Boschen Jun 7 '17 at 23:54
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    $\begingroup$ No I said bin 9 is 90 Hz and that is correct. It is also correct that bin[9] is -10 Hz. Look a the graphs of frequency that both i and Fat32 have posted and you can see that the frequencies from 0 to Fs are the same as 0 to Fs/2 and -Fs/2 to 0! So for a 10 pt FFT in a 100 Hz system, the frequencies are both 0, 10, 20, 30, 40, 50, 60, 70, 80, 90 AND equally 0, 10, 20, 30, 40, 50, -40, -30, -20, -10! Check out FFTSHIFT in Matlab as it does this translation. If this is still confusing, this would be a good different question to ask (or search if it has already been answered) $\endgroup$ – Dan Boschen Jun 8 '17 at 3:31
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The following figure also displays a simplified graphical view of the frequency normalization procedure as a result of the sampling of a continuous-time signal enter image description here

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