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An introduction to the frequency-domain and negative frequency
David Dorran Page 1
Table of Contents
How to use this series of documents .................................................................................... 2
An introduction to the frequency-domain ........................................................................... 3
What are sinusoidal waveforms? ...................................................................................... 5
All signals can be decomposed into sinusoidal waveforms ............................................. 7
A note on the duration of a sinusoidal waveform ............................................................ 8
A note on Fourier analysis ................................................................................................ 9
How to use Octave/Matlab’s fft function ............................................................................. 9
Using Octave/Matlab’s fft function to analyse a bass guitar signal ................................ 9
Interpreting the output of the fft function ...................................................................... 13
Negative Frequencies ........................................................................................................... 16
Visualising complex exponentials and negative frequency ............................................ 17
Why complex exponential waveforms have a helix shape............................................. 20
Another perspective on complex exponentials .............................................................. 24
The magnitude spectrum of a complex exponential ..................................................... 26
An introduction to the frequency-domain and negative frequency
David Dorran Page 2
How to use this series of documents
“Digital Signal Processing Foundations” provides a gentle introduction to the world of
DSP. This series of documents deals with topics relevant to DSP and they provide links to
online video content in an effort to make some perhaps tricky concepts easier to
understand for the reader.
If you are completely new to DSP then I’d recommend you take a look at the foundations
document first. After reading this document you may wish to review frequency response
in “Digital Filters – A practical guide”.
I believe that the approach of integrating text with video takes advantage of the unique
visualisations that video material has to offer with the more in-depth detail and speed of
review that text-based material does. Most of the video material relates to my own
youtube channel, which at the time of writing (2023) had over 18,000 subscribers and 3.7
million views.
I also provide Octave/Matlab code examples throughout the document and I’d encourage
anyone who wants to develop practical DSP skills to download Octave, which is available
free of charge, and implement your ideas.
My intention is to continue this series so as to deal with the major elements of DSP, such
as convolution, correlation, the Z-Transform and so on. I’ll post updates to this series at
http://www.pzdsp.com/docs if you want to check out any new additions.
If you find any of this work useful I’d be most grateful if you could cite the relevant
resource when appropriate to provide recognition.
Regards,
David
An introduction to the frequency-domain and negative frequency
David Dorran Page 3
An introduction to the frequency-domain
When someone plays the guitar different sounds are created because the guitar strings
vibrate or oscillate at different frequencies. A similar effect can be heard if you stretch an
elastic band between your fingers and pluck it and you’d notice that changing either the
length or the tension of the band would alter the frequency of the sound since this causes
the band to vibrate at a different rate or frequency.
When something is oscillating a repeating pattern is being produced over time. This can
be seen with a vibrating elastic band as it moves backwards and forwards through its
initial position.
The repeating nature associated with the movement of a guitar string can also be seen in
a plot of the audio signal it produces, as shown below, where the amplitude of a bass guitar
audio signal is seen to move up and down over time as the strings vibrate. You should
note that the rate of oscillation of the string is the same as the rate of oscillation of the
audio signal since it is the string vibrations that cause pressure variations in the air which
we perceive as sound (The audio recording of the bass guitar signal shown above can be
downloaded from pzdsp.com/sig1). The change in air pressure can also be recorded by a
microphone and stored on a computer as a discrete signal i.e. a sequence of numbers that
were obtained by measuring the sound pressure at regular time intervals.
The frequency-domain representation of a signal is a convenient way of showing the
oscillation rate associated with a signal, as explained in the following paragraph.
From the figure above, the sound pressure oscillates after the initial ‘attack’ or transient
component at the start of the signal. This plot of pressure variation over time is referred
to as a time-domain plot and by looking closely at this plot you can see that the time to
200ms segment
Time-domain view of a
1.5 second recording of a bass guitar
Magnitude
Frequency Hz
Frequency-domain view
time
Amplitude
50 100 150 200 250 300
Fundamental (55 Hz)
An introduction to the frequency-domain and negative frequency
David Dorran Page 4
complete one cycle of an oscillation is about 1.82 milliseconds (approx. 11 cycles over a
200ms segment). In other words, the cycle is repeating about 55 times every second. To
the right of the time-domain plot is a plot of the magnitude spectrum which is a frequency-
domain representation that can be used to quickly determine the rate of oscillations in
time-domain signals. The three relatively large ‘spikes’ shown in the magnitude spectrum
represent the fundamental frequency (55 Hz) and the first two harmonics (110 Hz and
165 Hz). You should notice that you can tell the rate of oscillation (55 Hz) quite easily
when you look at the signal in the frequency-domain; much more quickly and easily than
by analysing the period of the time-domain signal.
This type of repeating pattern doesn’t just happen with audio signals and it can be
observed in many signals, including those from our heart. Your heart will beat at
particular rate, or frequency, depending on what you are doing and your heart rate will
increase if you go for a run or cycle. Engineers and scientists (and musicians and doctors!)
are often analysing the repeating nature of signals and the frequency-domain view of a
signal shows the frequency of the repeating patterns in a convenient graph.
The frequency-domain view of a signal provides another way of analysing a signal which
can provide valuable insight into a signals’ behaviour. I find it useful to relate this to the
way an architect has different drawings of a building depending on who she is dealing
with: A client would find it easier to visualise what the building would look like by
examining a 3-D view of the building; while a builder would require detailed plans in order
to construct the building. Both sets of drawings are representations of the same building
and both have their uses. It’s the same with the time-domain
and frequency-domain views of signals – both represent the
same signal and both can be very useful when analysing
signals. Here’s a link to a video which demonstrates the
benefit of both the time-domain view and frequency-domain
view of a signal pzdsp.com/vid12.
0123456789 123456
Magnitude
Frequency (Hz)
0
Ampliitude
Time (Seconds)
Fundamental (1.1 Hz)
Time-domain view of an ECG signal Frequency-domain view of the ECG Signal
Magnitude Spectrum
An introduction to the frequency-domain and negative frequency
David Dorran Page 5
Frequency-domain graphs of signals are very easy to create using software tools like
Octave and Matlab and they make use of Fourier analysis techniques to extract frequency
information from a time-domain signal (more on this later!). The basic principle behind
all of the Fourier analysis techniques is that any signal can be broken down into a set of
sinusoidal signals and this concept is explored further in the next couple of subsections.
What are sinusoidal waveforms?
A sinusoidal waveform that oscillates smoothly over time (see the plot below) and is
associated with many signals that occur in nature. For example, when you whistle you
create pressure variations in the air which have a sinusoidal shape or of you were to allow
an object attached to the end of a spring bounce up and down then the motion of the
object would also be sinusoidal (see pzdsp.com/vid13). Even more interestingly it turns
out that sinusoidal waveforms are a fundamental building block of any signal so it’s worth
spending some time getting used to what they look like and how they can be represented
mathematically. This fact was shown mathematically by a French mathematician called
Jean Baptiste Joseph Fourier (1768-1830).
There are three features of sinusoidal waveforms that you’ll need to be comfortable with
to fully appreciate Fourier analysis: frequency, amplitude and phase offset.
The figure above shows a time-domain plot of a cosine waveform to the left and its
corresponding magnitude spectrum to the right. From the time-domain view notice
that the sinusoids amplitude oscillates between 1.5 and -1.5 which means that the
amplitude of the sinusoid is 1.5. You’ll notice that the sinusoid is repeating every 0.5
seconds, in other words it has a period of 0.5 seconds, which means that it has a frequency
of 2 Hz. I’d recommend you check out the interactive animation at pzdsp.com/sinusoids
in order to get a clearer idea about these parameters.
A sinusoidal waveform of
amplitude 1.5 and frequency 2 Hz
0 0.5 1 1.5 2 2.5 3
-1
0
1
Amplitude
time (seconds)
Magnitude
Frequency (Hz)
Frequency-domain view
12345 6
Time-domain view
1.5
An introduction to the frequency-domain and negative frequency
David Dorran Page 6
The frequency-domain plot of the sinusoid above shows a single
‘spike’ at a frequency of 2 Hz. Anytime you have a time-domain
plot of a single sinusoid you will observe a single ‘spike’ in the
frequency-domain and the position of the ‘spike’ on the frequency
axis corresponds to the frequency of the sinusoid. The magnitude
(height) of the ‘spike’ is proportional to the amplitude of the
sinusoid. You’ll see examples of signals with more than one
sinusoid present in the next section.
Before we look at the phase associated with this sinusoid lets first
take a look at a mathematical function often used to represent a
sinusoid which is shown below:
()=cos(2 +)
The A parameter specifies the amplitude of the sinusoid; f specifies the frequency and
(Greek letter phi) parameter specifies the phase offset (also referred to as the initial phase
or phase). The t variable represents time and the mathematical expression is evaluated for
a range of values of t in order to create a time-domain signal. So, if you wanted to recreate
the plot of the sinusoid shown above you’d substitute A with 1.5, f with 2 and with 0 to
give x(t) = 1.5cos(4πt), and then you could evaluate this for a number of values of t before
finally plotting your graph of x(t) against time.
You should notice that when the phase value is zero that the waveform will be a maximum
when t=0 and every period of the waveform after that. Changing the phase will change the
times when the maximum of the sinusoid will occur. You should try this out for yourself
using the code above and you should also observe that adding 2π to any phase offset value
you try out will produce the exact same waveform. For example, the waveform produced
when the phase offset is set to 1.4 will be the same as the waveform produced when the
Octave code to create a
plot of a sinusoid:
A = 1.5;
f = 2;
phi = 0;
duration = 1; %1 second
T = 1/f;
t=0:T/100:duration;
x = A*cos(2*pi*f*t + phi);
plot(t,x)
xlabel(‘time (seconds)’)
ylabel(‘Amplitude’)
An introduction to the frequency-domain and negative frequency
David Dorran Page 7
phase is set to 1.4+2π, or 1.4+4π, or even 1.4-2π for that matter. In fact, you will find that
for any integer k the following relationship holds:
cos(2+)=cos(2++2)
All signals can be decomposed into sinusoidal waveforms
The French mathematician Jean-Baptiste Joseph Fourier showed that any signal can be
recreated by adding sinusoidal signals together. (See pzdsp.com/vid14 and
pzdsp.com/vid15 for video tutorials/demonstrations on this concept).
The frequency-domain view of a signal provides a way to visualise the sinusoids that make
up a signal i.e. the sinusoids that when added together reproduce the original signal. The
magnitude spectrum shows the amplitudes of the various sinusoids which make up a
signal, while the phase spectrum shows the phases of the sinusoids which make up a
signal.
The figure above shows a waveform (top) which is a plot of the time-domain signal
produced when the two sinusoids shown below it are added together. The frequency-
domain view of this signal contains two spikes; the spike at 2 Hz is larger than the one at
24 Hz because the 2 Hz sinusoid is larger (5 times larger) than the 24 Hz sinusoid.
An introduction to the frequency-domain and negative frequency
David Dorran Page 8
The figure to the right shows the
magnitude spectrum of a signal in the
bottom plot; with the time-domain
view of the same signal shown in the
top plot. Each of the 'spikes' in the
magnitude spectrum represents a
sinusoid (there are 4 in total
indicating the presence of 4 sinusoids
in the signal; in other words the signal
could be reproduced by adding four
sinusoids together). Each of the four
sinusoids, which when summed
together produce the time-domain signal shown in the top plot, are shown in the middle
plot. The green sinusoid has 5 cycles over the one second duration of the segment shown
and therefore has a frequency of 5 Hz; it has the largest amplitude, as can also be seen in
the corresponding magnitude spectrum plot where the ‘spike’ shown at 5 Hz is the largest.
It can also be seen in the magnitude spectrum that the ‘spike’ at 8 Hz is less than half the
height of the 5 Hz spike; this can also be seen in the middle plot whereby the sinusoid
with 8 cycles in one second has an amplitude of less than a half the amplitude of the 5 Hz
sinusoid.
The phase values for each of the sinusoids present in the signal are 0, 0, 3.14, 2.13 radians
for the 1, 5, 8, and 10 Hz components. These phase values are phase shifts relative to cosine
waveforms. A plot of the phase spectrum shows the phase values plotted against frequency
in a similar way to the magnitude spectrum showing the magnitude values plotted against
frequency.
If you would like to see a practical application of the frequency-domain then take a look
at pzdsp.com/vid12.
A note on the duration of a sinusoidal waveform
From the mathematical description of a sinusoid a sinusoidal waveform exists for all
instances of time. In this document I show plots of sinusoidal segments which have a finite
duration and you’ll notice that I still refer to these plots of sinusoidal segments as
sinusoids, which is, strictly speaking, incorrect but makes the document a bit easier to
read.
An introduction to the frequency-domain and negative frequency
David Dorran Page 9
A note on Fourier analysis
Fourier transforms are mathematical techniques which determine the sinusoidal
parameters (amplitude, frequency and phase) of the sinusoidal waveforms that are
present in a mathematical function. When working with ‘real-world’ data (as opposed to
mathematical functions) the discrete Fourier transform (DFT) is used determine the
frequency-domain characteristics of this ‘real world’ data/signals. A detailed description
of how the DFT works is provided in the document entitled “The Discrete Fourier
Transform - A practical approach” available at https://pzdsp.com/docs.
This document doesn’t provide examples of Fourier transforms and the interested reader
should explore other resources for further insight. From my experience DSP practitioners
will gain a more valuable insight from understanding the DFT first as it can be used on
practical signals. It can be easy to get so caught up in the mathematics of the Fourier
transform that you can forget what the purpose of it is!
How to use Octave/Matlab’s fft function
The fft function can be used to determine the amplitude, frequencies and phases of the
sinusoids that a signal is comprised of and is frequently used to obtain a frequency-
domain plot of a signal. In this section I’ll explain how to use the fft function without
getting into detail on its inner workings. You should note that the fft function is an
implementation of the Discrete Fourier Transform algorithm which is described in detail
in the document “The Discrete Fourier Transform - A practical approach” available at
https://pzdsp.com/docs.
In this section I’ll first show how to create a frequency-domain plot of the bass guitar
signal used in the Introduction, then I’ll provide another example which provides more
insight on how to use the fft to analyse a signal which is based on the popular video on
the subject that I created in 2012 pzdsp.com/vid16.
Using Octave/Matlab’s fft function to analyse a bass guitar signal
The following code can be used to load in an audio signal and plot its frequency content.
The audio file in the example can be downloaded from pzdsp.com/sig1 and you should
make sure the audio file is stored/saved in the ‘present working directory’ – this can be
determined by typing pwd at the command line.
An introduction to the frequency-domain and negative frequency
David Dorran Page 10
>> [b fs]= audioread('bass_note.wav'); % the variable b
contains the audio samples. The audioread function also
returns the sampling rate,fs, which is 44100 in this case
>> B = fft(b); % the fft returns 67822 complex numbers
which are stored in the array variable B. Note that, by
convention, capital letters are used to store frequency-
domain information while lowercase are used for time-
domain. There are 67822 samples in the time-domain signal
b. The fft function returns the same number of values as
are in the signal being analysed i.e. the time-domain
signal b in this case.
>> B_mags = abs(B); % the abs function determines the
magnitudes of the 67822 complex numbers.
The second line in the code above is the one that does all the hard work. The fft function
analyses the time-domain signal b (as described in detail in the next section) to determine
the magnitudes and phases of the sinusoids required to reproduce the time-domain signal
b.
One of the most common ways to visually analyse the frequency content of signal is to
plot the magnitudes of the values returned by the fft function, which provides a plot of
the magnitude spectrum. You should note that there are numerous ways to plot the
magnitude spectrum and the video available at pzdsp.com/vid17 provides a detailed
explanation on how to do so. The following code shows one common method of plotting
the magnitude spectrum against frequency using units of hertz, where the xlim([0 500])
command limits the range of frequencies being displayed to be from 0 to 500 Hz.
>> plot([0:length(b)-1]/length(b)*fs , B_mags)
>> xlim([0 500]); %limit the range of frequencies to be
from 0 to 500 Hz
>> xlabel('Frequency (Hz) '); ylabel('Magnitude');
A more detailed explanation of this code can be found from pzdsp.com/vid17. This code
will produce the following plot in which the fundamental and first two harmonics of the
guitar note can be clearly seen. There is a fundamental frequency component at about
55Hz, with strongly present harmonics at 110 Hz and 155 Hz, as indicated by the large
spikes at these frequencies
An introduction to the frequency-domain and negative frequency
David Dorran Page 11
The time-domain view of this signal can be plotted using the following code, and you can
see that the signal contains a sharp attack/transient element from when the bass guitar
string was plucked.
>> [b fs]= audioread('bass_note.wav');
>> t = [0:length(b)-1]*1/fs;
>> plot(t,b)
>> xlabel('Time (seconds)'); ylabel('Amplitude')
The figure above also highlights a steady-state region in which the signal is reasonably
stationary. This steady-state/stationary segment can be reproduced reasonably well by
adding just three sinusoidal components, as shown by the code and plots below. Note that
the code doesn’t show how the amplitudes, frequencies and phases of the three sinusoidal
components are determined.
>> [ip fs]= audioread('bass_note.wav');
>> N = 9670;
050 100 150 200 250 300 350 400 450 500
Frequency (Hz)
0
1000
2000
3000
4000
Magnitude
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Time (seconds)
-0.4
-0.2
0
0.2
0.4
Amplitude
Attack/Transient
A steady-state region
An introduction to the frequency-domain and negative frequency
David Dorran Page 12
>> stationary_seg = ip(10000:10000+N-1);
>> t = [0:N-1]/fs; t_offset = 10000/fs;
>> subplot(2,2,1); plot(t+t_offset, stationary_seg)
>> title('Original segment')
>> xlabel('Time (seconds)'); ylabel('Amplitude');
>> subplot(2,2,2);
>> plot([0:N-1]/N*fs , abs(fft(stationary_seg)))
>> xlim([0 250]); %limit frequencies from 0 to 250 Hz
>> xlabel('Frequency (Hz) '); ylabel('Magnitude');
>> title('Magnitude Spectrum')
>> fundamental = 0.112*cos(2*pi*54.7*t-0.82);
>> harmonic1 = 0.072*cos(2*pi*109.7*t+3);
>> harmonic2 = 0.028*cos(2*pi*164.6*t + 0.83);
>> synth_sig = fundamental + harmonic1 + harmonic2 ;
>> subplot(2,2,3);plot(t+t_offset, synth_sig);
>> xlabel('Time (seconds)'); ylabel('Amplitude');
>> title('Synthesised segment')
>> subplot(2,2,4); plot(t+t_offset, fundamental);
>> hold on ; plot(t+t_offset, harmonic1);
>> plot(t+t_offset, harmonic2);
>> xlabel('Time (seconds)'); ylabel('Amplitude');
>> title('Three synthesis sinusoids')
0.25 0.3 0.35 0.4
Tim e (sec onds)
-0.2
-0.1
0
0.1
0.2
Amplitude
Original segme nt
050 100 150 200 250
Frequency (Hz)
0
200
400
600
Magnitude
Magnitude Spectrum
0.25 0.3 0.35 0.4
Tim e (sec onds)
-0.2
-0.1
0
0.1
0.2
Amplitude
Synthesised segme nt
0.25 0.3 0.35 0.4
Tim e (sec onds)
-0.2
-0.1
0
0.1
0.2
Amplitude
Three synthesis sinusoids
An introduction to the frequency-domain and negative frequency
David Dorran Page 13
Interpreting the output of the fft function
The fft function returns a sequence of complex numbers, and these complex numbers
describe the amplitude and phases of the sinusoidal waveforms that a time-domain signal
is comprised of. In this section I’ll attempt to explain how to interpret the complex
numbers returned by the fft function using some synthesised example signals, as
explained in pzdsp.com/vid16.
Let’s start by synthesising a time-domain signal that contains three sinusoids. The fft
function should be able to determine the amplitudes and phases of these three sinusoids
so let’s see how it does it.
>> fs = 1000;
>> t = 0 : 1/fs :1.5-1/fs;
>> x = 3*cos(2*pi*20*t + 0.2) + 1*cos(2*pi*30*t -0.3) +
2*cos(2*pi*40*t + 2.4);
>> plot(t,x);
>> xlabel('Time (seconds) ');
>> ylabel('Amplitude');
We know that the time-domain signal shown above contains three sinusoids of
frequencies 20 Hz, 30 Hz and 40 Hz with phase offsets of 0.2 radians, -0.3 radians and 2.4
radians, respectively, and amplitudes 3, 1, and 2, respectively. The time-domain signal
contains 1500 samples (sampling rate is 1000 Hz) and when we apply the fft function to
this signal 1500 complex numbers are returned. If we plot the magnitudes of these 1500
complex numbers, as shown below, we can see three ‘spikes’ on the left hand side of the
plot with another three spikes ‘mirrored’ on the right hand side. The three pairs of ‘spikes’
represent the three sinusoidal components that the original signal is comprised of.
>> X = fft(x);
0 0.5 1 1.5
Tim e (seconds)
-4
-2
0
2
4
Am plitude
An introduction to the frequency-domain and negative frequency
David Dorran Page 14
>> plot(abs(X)); xlabel('Frequency (bins)');
ylabel('Magnitude');
>> title('Magnitude Spectrum')
The amplitude of the ‘spikes’ correspond to the amplitude of the sinusoids. Referring to
the three ‘spikes’ on the left-hand side; the spike furthest to the left corresponds to the 20
Hz sinusoid which has the largest amplitude; the middle ‘spike’ has the lowest amplitude
and corresponds to the 30 Hz sinusoid; while the ‘spike’ to the right of the grouping is
twice the amplitude of the middle ‘spike’ and corresponds to the sinusoid with a frequency
of 40 Hz.
If we took a closer look at the values of the variable X we’d see that they are complex
numbers that contain a lot of zero values. While showing all 1500 values is impractical we
can use the following matlab code to look at a few:
>> X(30:32) % X(30) X(31) X(32)
0+0j 2205.15+447j 0+0j
>> X(45:47) % X(45) X(46) X(47)
0+0j 716.5-221.64j 0+0j
>> X(60:62) % X(60) X(61) X(62)
0+0j -1106.09+1013.19j 0+0j
We can see that there are three non-zero values at indices 31, 46 and 61. The magnitudes
of these values are 2250, 750 and 1500, respectively, and these values can also be
determined from the plot of the magnitude spectrum by examining the amplitude of each
of the ‘spikes’. This should make sense, since the plot of the magnitude spectrum is simply
a visual representation of the magnitudes of the variable X. It’s worth noting that if we
divide these magnitude values by 750 (which is half the number of values in X) then we
0500 1000 1500
Frequency (bins)
0
500
1000
1500
2000
2500
Magnitude
Magnitude Spectrum
An introduction to the frequency-domain and negative frequency
David Dorran Page 15
get a result of 3, 1 and 2 which exactly match the amplitudes of the three sinusoids that
the synthesised signal is comprised of.
The phase angles of the complex numbers of X at indices 31, 46 and 61 are 0.2 radians, -0.3
radians and 2.4 radians, respectively. By referring to the code which synthesised the time-
domain signal we can see that these phase angles directly correspond to the phases of the
sinusoids that the synthesised signal is comprised of.
So, we can see that the fft function can determine the amplitudes and phase of the
sinusoids that a signal is comprised. The remaining piece of information is the frequency
of each of those sinusoids and to determine the frequency we have to examine the indices
of the non-zero values of X i.e. 31, 46 and 61. Before continuing its important to note that
matlab and octave index the first value of an array with the number 1, while
mathematicians (and most other programming) languages will index the first element of
an array with the number 0. The values of the array returned by the fft function (stored in
the variable X) are referred to as bin values, with the first element of the array being
referred to as bin number 0. The non-zero values of the variable X occurring at indices 31,
46 and 61 therefore correspond to bin numbers 30, 45 and 60. These bin numbers are
related to frequency with the bin numbers associated with the left-hand side of the
magnitude spectrum being converted to frequency in hertz using the following formula:
f = k.fs/N
where f is the frequency associated with bin k, fs is the sampling frequency and N is the
total number of bins (which is equal to the number of values in the variable X).
Using this formula for bin values k set to 30, 45 and 60 gives frequencies 20 Hz, 30 Hz and
40 Hz, which correspond to the frequencies of the sinusoids used to synthesise the time-
domain signal.
An introduction to the frequency-domain and negative frequency
David Dorran Page 16
Negative Frequencies
By this stage you should be comfortable with the idea that all signals can be decomposed
into sinusoidal waveforms. However, a sinusoidal waveform can also be considered as
being the sum of two other waveforms, known as complex exponentials (explained in
detail later).
When you plot the magnitude spectrum of a signal you’ll see that the spectrum contains
‘mirrored’ components/spikes at each end of the spectrum. For example, earlier on, when
we examined the signal which comprised of three sinusoids, the following plot was
produced (the plot has been modified so that frequency components are different
colours). Notice how the left-hand side of the spectrum is a ‘mirror image’ of the right-
hand side.
The three components/spikes to the left are associated with what are referred to as
‘positive frequency complex exponentials’ while the ones to the right are associated with
‘negative frequency complex exponentials’. The plot shows ‘mirrored pairs’ of frequency
components using three different colours (red, green yellow). Each ‘mirrored pair’ of
complex exponentials actually represent a single sinusoidal waveform, as shown
mathematically below. In most situations the ‘mirrored’ half of the spectrum is not
required for engineers and scientists to analyse the magnitude spectrum, since they are
effectively redundant, and so a single-sided spectrum is often displayed for simplicity.
Note that you will often find the magnitude spectrum plotted in units of Hertz with the
negative frequency components shown to the left, as shown below. Octave/Matlab code
to create these plots is explained in pzdsp.com/vid17.
0500 1000 1500
Frequency (bins)
0
500
1000
1500
2000
2500
Magnitude
Magnitude Spectrum
Three positive frequency
components
Three negative frequency
components
An introduction to the frequency-domain and negative frequency
David Dorran Page 17
Complex exponentials are described mathematically by the following function:
()=()
The A parameter specifies the amplitude of the sinusoid; f specifies the frequency and
parameter specifies the phase offset. Notice the similarity of these parameters with
sinusoidal parameters.
Using the well-known Euler’s Formula e jθ = cos(θ) + jsin(θ), and the fact that cos(-θ) = cos(θ)
and sin(-θ) = -sin(θ), it can be shown that e -jθ = cos(θ) - jsin(θ). It therefore can also be shown
that
cos()= +
2
Using this result it can therefore be shown that any sinusoidal waveform is the sum of two
complex exponential waveforms since
cos(2 +)=()+()
2
The complex exponential () is the ‘positive frequency component’ while
() is the ‘negative frequency component’.
Visualising complex exponentials and negative frequency
Complex exponentials have a helix shape (like a spring or corkscrew) and they can be
tricky to visualise if you’re not sure what you’re looking at. To make this process easier I’d
like you to show helix examples that are reasonably easy to interpret. I’d also like you to
appreciate that the two examples below spiral or rotate in opposite directions. The helix
-500 -400 -300 -200 -100 0100 200 300 400 500
Frequency (Hz)
0
500
1000
1500
2000
2500
Magnitude
Magnitude Spectrum
Three negative frequency
components
Three positive frequency
components
An introduction to the frequency-domain and negative frequency
David Dorran Page 18
to the left rotates in an anti-clockwise direction while the one to the right rotates in a
clockwise direction. This direction of rotation is extremely relevant for understanding
negative frequency.
The direction of rotation of the above example helixes are relatively easy to interpret
because of their 3-dimensional structure and colouring. The helix shapes you’ll see later
may be more difficult as they will not be rendered in such a ‘solid’ format. However, once
you appreciate that you are viewing a helix this should not be an issue for most readers.
The plot below shows another helix which rotates in a clockwise direction. The helix to
the right is the same helix but viewed ‘face on’ to highlight the fact that the helix has a
circular shape when viewed from that perspective.
Now let’s take a look at a visualisation of a complex exponential waveform (the colours on
the helix are there just to make it easier to see the 3-D shape!). I’ll explain where this
visualisation comes from later, but it can be useful to have a mental picture before getting
into the deeper explanation. Don’t be worried if you don’t fully understand the illustration
at this stage! You’ll notice that there are three axes, a time axis, a real axis and an imaginary
axis. The complex exponential waveform shown in the plot is a helix (spring-like shape)
that rotates in an anti-clockwise direction. It is the direction of rotation that is key to
understanding negative frequencies!
An introduction to the frequency-domain and negative frequency
David Dorran Page 19
The complex exponential waveform above is associated with a positive frequency because
it rotates in an anti-clockwise direction. I’ll explain why this is case later on, but for the
moment just accept it, and also accept that negative frequencies are associated with
complex exponential waveforms that rotate in a clockwise direction, like the one
illustrated below.
-5
-4
-3
-2
5
-1
0
1
2
3
4
5
6
04
2
-5 0
-5
-4
5
-3
-2
-1
0
6
1
2
0
3
4
4
5
2
-5 0
An introduction to the frequency-domain and negative frequency
David Dorran Page 20
As is the case with sinusoidal waveforms, complex exponential waveforms have an
amplitude, frequency and phase associated with them, however the direction of rotation
introduces another feature i.e. a positive frequency (for complex exponentials that rotate
anti-clockwise) and negative frequency (for complex exponentials that rotate clockwise).
Please note that a persistent feature of complex exponentials is that they have a helix
shape. The negative frequency waveform shown has an amplitude of 3 while the positive
frequency waveform has an amplitude of 2. Also, notice that the negative frequency
complex exponential has four rotations over 6 second i.e. a frequency of 0.66 Hz. The
positive frequency complex exponential has three complete rotations over 6 seconds i.e.
a frequency of 0.5 Hz.
At this stage you have been exposed to the most important aspects of complex
exponentials and negative frequencies. If you are able to appreciate the idea that positive
and negative frequency relates to the direction of rotation of the helix shapes associated
with complex exponentials then you are in a very good position!
Why complex exponential waveforms have a helix shape
Euler’s Formula states that e jθ = cos(θ) + jsin(θ). Notice the presence of the imaginary unit
=1 in the formula!
Complex exponentials are therefore the sum of a cosine waveform and a sine waveform,
where the sine waveform is multiplied by the imaginary unit =1, i.e.
()=cos(2 +)+ sin(2 +)
To create a plot of a sinusoidal waveform you could evaluate the mathematical expression
cos(2 +) for a range of values of the time variable t and plot the resulting sequence
of ‘real’ amplitude values against time. If you were to evaluate complex exponential
waveforms (()) for a range of values of the time variable t, then you would obtain
a sequence of ‘complex’ values i.e. values with ‘real’ and ‘imaginary’ terms.
Since complex exponentials have ‘complex terms’ (i.e. real and imaginary terms) then we
can use an argand diagram to visualise these terms. The following video may help some
readers at this stage pzdsp.com/vid27.
Let’s evaluate a complex exponential waveform which has unit amplitude, zero
phase offset and frequency 1 Hz, just to see the shape that will be produced (and yes, it
An introduction to the frequency-domain and negative frequency
David Dorran Page 21
will be a helix!). The table below shows the result of evaluating =cos(2)+
sin(2) , for a range of increasing values of t.
time (t)
()
()
=()+()
0 1.0000 0.000 1.0000 + 0.0000j
0.2 0.3090 0.9511 0.3090 + 0.9511j
0.4 -0.8090 0.5878 -0.8090 + 0.5878j
0.6 -0.8090 -0.5878 -0.8090 - 0.5878j
0.8 0.3090 -0.9511 0.3090 - 0.9511j
Plotting these complex numbers on an argand diagram shows that they all lie on a unit
circle. Notice how the complex numbers evolve in anti-clockwise direction as time
increases.
If we add a time axis to the argand plot the helix shape will emerge and you should notice
that the helix rotates in an anti-clockwise direction. The red line showing the shape of the
helix is the waveform evaluated over the range 0 to 2 seconds, at intervals of 0.001
seconds. These 2000 values were, of course, determined using a computer!
An introduction to the frequency-domain and negative frequency
David Dorran Page 22
Now let’s evaluate the negative frequency complex exponential waveform , over
increasing values of t. These complex numbers evolve as helix shape rotating in a
clockwise direction as time increases, for this case.
time (t)
()
()
=() ()
0
1.0000
0.000
1.0000 + 0.0000j
0.2
0.3090
0.9511
0.3090 - 0.9511
j
0.4
-0.8090
0.5878
-0.8090 - 0.5878j
0.6
-0.8090
-0.5878
-0.8090 + 0.5878j
0.8
0.3090
-0.9511
0.3090 + 0.9511
j
-1
-0.5
0
0.5
-1
1
02
1.5
1
0.5
10
-1
-0.5
-1
0
0.5
1
02
1.5
1
10.5
0
An introduction to the frequency-domain and negative frequency
David Dorran Page 23
From above you can see that the positive frequency complex exponential rotates in
an anti-clockwise direction while the negative frequency complex exponential
rotates in a clockwise direction.
You should appreciate that in general, a complex exponential waveform will always have
a helix shape. The waveform given by () will rotate at rotations per second,
and that the radius of the helix will be . The phase offset simply moves the ‘starting
point’ of the helix at time t = 0. The same applies to a waveform given by (), the
only difference is the direction of rotation of the helix shape.
The following Matlab/Octave code can be used to create plots of helix’s that the interested
reader may find beneficial. As always skip the code provided if it’s not helpful.
>> T = 0.001; % make this variable smaller to produce smoother
plots
>> t = 0: T : 3; % used to evaluate the waveform from t=0 to t=3
(in steps of T)
>> A = 0.7; f = 2; w=2*pi*f; phi = -pi/2;
>> freq_direction = 1; %set to 1 for positive frequency
>> x = A*exp(freq_direction*j*(w*t + phi)); %evaluate the
complex exponential waveform for the times specified earlier
>> plot3(real(x), t,imag(x))
>> ylabel('Time axis (seconds)')
>> xlabel('Real axis')
>> zlabel('Imaginary axis')
>> grid on
>> set(gca,'CameraPosition',[8.9444 -18.4657 6.5444])
3
-1
2
-1
-0.5
Time axis (seconds)
0
-0.5
Imaginary axis
Real axis
0.5
1
0
1
0.5 0
1
An introduction to the frequency-domain and negative frequency
David Dorran Page 24
Another perspective on complex exponentials
If you feel you have a good understanding of complex exponentials and negative frequency
then feel free to skip this section. I just wanted to provide a slightly different perspective
into visualising complex exponentials that might help some readers.
Complex exponentials are the combination of a cosine waveform and sine waveform.
Consider the plot below which shows a sine waveform and a cosine waveform. In addition,
there is a plot showing 5 points (coloured black, greed, red, blue, and purple) which all
appear on the circumference of a circle.
There are also 5 points shown on the cosine and sine waveforms and they appear in order
of black, red, green, blue, and purple along the time axis, at the same time locations.
Notice how the horizontal position of the black point on the circle corresponds to the
amplitude of the black point shown on the cosine waveform, and that its vertical position
corresponds to the amplitude of black point on the sine waveform.
Also notice how the horizontal position of the green point on the circle corresponds to
the amplitude of the green point shown on the cosine waveform, and that its vertical
position corresponds to the amplitude of green point on the sine waveform.
An introduction to the frequency-domain and negative frequency
David Dorran Page 25
The position of all the different coloured points on the circle are determined in a similar
way. The helix plot to the right above shows that these points on the circle form a helix
shape as they evolve over time.
The plot below is very similar to the previous plot except that the sine waveform has been
inverted. Inverting the sine waveform has the affect of causing the helix to rotate in the
opposite direction.
An introduction to the frequency-domain and negative frequency
David Dorran Page 26
The magnitude spectrum of a complex exponential
As discussed earlier, a Fourier analysis of a sinusoidal waveform will contain two
components in its magnitude spectrum i.e. a positive frequency complex exponential and
a negative frequency complex exponential.
As you might expect, a Fourier analysis of a complex exponential will contain a single
frequency component/’spike’ in the magnitude spectrum. A negative frequency complex
exponential will appear on the opposite side of the spectrum to a positive frequency
complex exponential (see Octave/Matlab code below for further insight).
The figure above shows the magnitude spectrum of a positive and negative complex
exponential in addition to the spectrum of a sinusoidal waveform. All waveforms have the
same amplitude, frequency and phase parameters (see code below). Notice that the two
frequency components associated with the sinusoidal waveform are half the magnitude
of the complex exponential waveforms. This makes sense since a sinusoidal waveform is
comprised of two complex exponentials which are half the amplitude of the sinusoid i.e.
cos(2 +)=()+()
2
050 100 150 200 250 300
frequency (bins)
0
100
200
300
Magnitude
Positive frequency complex exponential
050 100 150 200 250 300
frequency (bins)
0
100
200
300
Magnitude
Negative frequency complex exponential
050 100 150 200 250 300
frequency (bins)
0
100
200
300
Magnitude
Sinusoida l waveform
An introduction to the frequency-domain and negative frequency
David Dorran Page 27
>> T = 0.01; % make this variable smaller to produce time-domain
smoother plots
>> t = 0: T : 3-T; % used to evaluate the waveform from t=0 to t=3 (in
steps of T)
>> A = 0.8; f = 20; w=2*pi*f; phi = -pi/2;
>> pos_comp_exp = A*exp(j*(w*t + phi));
>> neg_comp_exp = A*exp(-j*(w*t + phi));
>> sinusoid = A*cos((w*t + phi));
>> subplot(3,1,1)
>> plot(abs(fft(pos_comp_exp)))
>> xlabel('frequency (bins)');ylabel('Magnitude');ylim([0 300]);
>> title('Positive frequency complex exponential')
>> subplot(3,1,2)
>> plot(abs(fft(neg_comp_exp)))
>> xlabel('frequency (bins)');ylabel('Magnitude');ylim([0 300]);
>> title('Negative frequency complex exponential')
>> subplot(3,1,3)
>> plot(abs(fft(sinusoid)))
>> xlabel('frequency (bins)');ylabel('Magnitude');ylim([0 300]);
>> title('Sinusoidal waveform')