3f3 – digital signal processing (dsp) - part1
DESCRIPTION
3F3 – Digital Signal Processing (DSP), January 2009, lecture slides 1, Dr Elena Punskaya, Cambridge University Engineering DepartmentTRANSCRIPT
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Introduction to Digital Signal Processing (DSP)
Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205
Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet,
Dr. Malcolm Macleod and Prof. Peter Rayner
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2
Books
Books:
– J.G. Proakis and D.G. Manolakis, Digital Signal Processing 3rd edition, Prentice-Hall.
– R.G Lyons, Understanding Digital Signal processing, 2nd edition, , Prentice-Hall.
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Digital: operating by the use of discrete signals to represent data in the form of numbers
Signal: a parameter (electrical quantity or effect) that can be varied in such a way as to convey information
Processing: a series operations performed according to programmed instructions
changing or analysing information which is measured as discrete sequences of numbers
What is Digital Signal Processing?
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Applications of DSP - Biomedical
Biomedical: analysis of biomedical signals, diagnosis, patient monitoring, preventive health care, artificial organs
Examples:
1) electrocardiogram (ECG) signal – provides doctor with information about the condition of the patient’s heart
2) electroencephalogram (EEG) signal – provides Information about the activity of the brain
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Applications of DSP - Speech
Speech applications:
Examples
1) noise reduction – reducing background noise in the sequence produced by a sensing device (microphone)
2) speech recognition – differentiating between various speech sounds
3) synthesis of artificial speech – text to speech systems for blind
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Applications of DSP - Communications
Communications:
Examples
1) telephony – transmission of information in digital form via telephone lines, modem technology, mobile phones
2) encoding and decoding of the information sent over a physical channel (to optimise transmission or to detect or correct errors in transmission)
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Applications of DSP - Radar
Radar and Sonar:
Examples
2) tracking
1) target detection – position and velocity estimation
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Applications of DSP – Image Processing
Image Processing:
Examples
1) content based image retrieval – browsing, searching and retrieving images from database
2) compression - reducing the redundancy in the image data to optimise transmission / storage
2) image enhancement
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Applications of DSP – Music
Music Applications:
Examples:
3) Manipulation (mixing, special effects)
2) Playback
1) Recording
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Applications of DSP - Multimedia
Multimedia:
generation storage and transmission of sound, still images, motion pictures
Examples: 1) digital TV
2) video conferencing
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DSP Implementation - Operations
To implement DSP we must be able to:
DSP Digital Signal
Digital Signal
1) perform numerical operations including, for example, additions, multiplications, data transfers and logical operations
either using computer or special-purpose hardware
Input
Output
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DSP chips
• Introduction of the microprocessor in the late 1970's and early 1980's meant DSP techniques could be used in a much wider range of applications.
DSP chip – a programmable device, with its own native instruction code
designed specifically to meet numerically-intensive requirements of DSP
capable of carrying out millions of floating point operations per second Bluetooth
headset Household appliances
Home theatre system
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DSP Implementation – Digital/Analog Conversion
DSP Digital Signal
Digital Signal
Reconstruction Analog Signal
2) convert the digital information, after being processed back to an analog signal
- involves digital-to-analog conversion & reconstruction (recall from 1B Signal and Data Analysis)
e.g. text-to-speech signal (characters are used to generate artificial sound)
To implement DSP we must be able to:
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DSP Implementation –Analog/Digital Conversion
DSP Digital Signal
Digital Signal
Sampling
Analog Signal
To implement DSP we must be able to:
3) convert analog signals into the digital information - sampling & involves analog-to-digital conversion (recall from 1B Signal and Data Analysis)
e.g. Touch-Tone system of telephone dialling (when button is pushed two sinusoid signals are generated (tones) and transmitted, a digital system determines the frequences and uniquely identifies the button – digital (1 to 12) output
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DSP Implementation
DSP Digital Signal
Digital Signal
Reconstruction Analog Signal Sampling
Analog Signal
To implement DSP we must be able to:
perform both A/D and D/A conversions
e.g. digital recording and playback of music (signal is sensed by microphones, amplified, converted to digital, processed, and converted back to analog to be played
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Limitations of DSP - Aliasing
Most signals are analog in nature, and have to be sampled loss of information • we only take samples of the signals at intervals and
don’t know what happens in between aliasing cannot distinguish between higher and lower frequencies
Sampling theorem: to avoid aliasing, sampling rate must be at least twice the maximum frequency component (`bandwidth’) of the signal
Gjendemsjø, A. Aliasing Applet, Connexions, http://cnx.org/content/m11448/1.14
(recall from 1B Signal and Data Analysis)
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• Sampling theorem says there is enough information to reconstruct the signal, which does not mean sampled signal looks like original one
Limitations of DSP - Antialias Filter
correct reconstruction is not just connecting samples with straight lines
needs antialias filter (to filter out all high frequency components before sampling) and the same for reconstruction – it does remove information though
Each sample is taken at a slightly earlier part of a cycle
(recall from 1B Signal and Data Analysis)
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Limitations of DSP – Frequency Resolution
Most signals are analog in nature, and have to be sampled loss of information
• we only take samples for a limited period of time
limited frequency resolution
does not pick up “relatively” slow changes
(recall from 1B Signal and Data Analysis)
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Limitations of DSP – Quantisation Error
Most signals are analog in nature, and have to be sampled loss of information • limited (by the number of bits available) precision in data
storage and arithmetic
quantisation error
smoothly varying signal represented by “stepped” waveform
(recall from 1B Signal and Data Analysis)
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Advantages of Digital over Analog Signal Processing
Why still do it?
• Digital system can be simply reprogrammed for other applications / ported to different hardware / duplicated
(Reconfiguring analog system means hadware redesign, testing, verification)
• DSP provides better control of accuracy requirements (Analog system depends on strict components tolerance, response may drift with
temperature) • Digital signals can be easily stored without deterioration (Analog signals are not easily transportable and often can’t be processed off-line)
• More sophisticated signal processing algorithms can be implemented
(Difficult to perform precise mathematical operations in analog form)
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Brief Review of Fourier Analysis
Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205
Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet,
Dr. Malcolm Macleod and Prof. Peter Rayner
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Time domain
Example: speech recognition
tiny segment
sound /a/ as in father
sound /i/ as in see
difficult to differentiate between different sounds in time domain
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How do we hear?
www.uptodate.com
Inner Ear
Cochlea – spiral of tissue with liquid and thousands of tiny hairs that gradually get smaller
Each hair is connected to the nerve
The longer hair resonate with lower frequencies, the shorter hair resonate with higher frequencies
Thus the time-domain air pressure signal is transformed into frequency spectrum, which is then processed by the brain
Our ear is a Natural Fourier Transform Analyser!
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Fourier’s Discovery
Jean Baptiste Fourier showed that any signal could be made up by adding together a series of pure tones (sine wave) of appropriate amplitude and phase
(Recall from 1A Maths)
Fourier Series for periodic square wave
infinitely large number of sine waves is required
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Fourier Transform
The Fourier transform is an equation to calculate the frequency, amplitude and phase of each sine wave needed to make up any given signal :
(recall from 1B Signal and Data Analysis)
1. The Discrete Time Fourier Transform (DTFT) of a sampled sequence{xn}, n = 0,±1,±2, . . . ,±! is given by
X (!) =!!
n="!
xne"jn!T .
2.
X (!) =
!"
"!
x(t)e"j!tdt.
where T is the sampling interval.
(a) Explain how the problems with computing the DTFT on a digitalcomputer can be overcome by choosing a uniformly spaced gripof N frequencies covering the range of ! from 0 to 2"/T andtruncating the summation in the expression above to just N datapoints; thus, determine the expression for the N -point DiscreteFourier Transform (DFT). [10%]
Xp =N"1!n=0
xne"j 2!N np, p = 0, . . . , N " 1
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp =N"1!n=0
xne"j 2!N np, p = 0, . . . , N " 1
(c) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =
1
1. The Discrete Time Fourier Transform (DTFT) of a sampled sequence{xn}, n = 0,±1,±2, . . . ,±! is given by
X (!) =!!
n="!
xne"jn!T .
2.
X (!) =
!"
"!
x(t)e"j!tdt.
where T is the sampling interval.
(a) Explain how the problems with computing the DTFT on a digitalcomputer can be overcome by choosing a uniformly spaced gripof N frequencies covering the range of ! from 0 to 2"/T andtruncating the summation in the expression above to just N datapoints; thus, determine the expression for the N -point DiscreteFourier Transform (DFT). [10%]
Xp =N"1!n=0
xne"j 2!N np, p = 0, . . . , N " 1
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp =N"1!n=0
xne"j 2!N np, p = 0, . . . , N " 1
(c) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =
1
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Prism Analogy
Analogy:
a prism which splits white light into a spectrum of colors
white light consists of all frequencies mixed together
the prism breaks them apart so we can see the separate frequencies
White light
Spectrum of colours
Fourier Transform
Signal Spectrum
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Signal Spectrum
Every signal has a frequency spectrum. • the signal defines the spectrum • the spectrum defines the signal
We can move back and forth between the time domain and the frequency domain without losing information
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Time domain / Frequency domain
• Some signals are easier to visualise in the frequency domain
• Some signals are easier to visualise in the time domain
• Some signals are easier to define in the time domain (amount of information needed)
• Some signals are easier to define in the frequency domain (amount of information needed)
Fourier Transform is most useful tool for DSP
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Fourier Transforms Examples
peaks correspond to the resonances of the vocal tract shape
they can be used to differentiate between sounds
in logarithmis units of dB
sound /i/ as in see
signal spectrum
cosine
added higher frequency component
sound /a/ as in father
in logarithmis units of dB
t
t
t
t
ω
Back to our sound recognition problem:
ω
ω
ω
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Discrete Time Fourier Transform (DTFT)
What about sampled signal?
The DTFT is defined as the Fourier transform of the sampled signal. Define the sampled signal in the usual way:
Take Fourier transform directly
using the “sifting property of the δ-function to reach the last line
1. The Discrete Time Fourier Transform (DTFT) of a sampled sequence{xn}, n = 0,±1,±2, . . . ,±! is given by
X (!) =!!
n="!
xne"jn!T .
2.
X (!) =
!"
"!
x(t)e"j!tdt.
3.
xs(t) = x(t)!!
n="!
"(t " nT )
4.
Xs (!) =
!"
"!
xs(t)e"j!tdt
5.
Xs (!) =
!"
"!
x(t)!!
n="!
"(t " nT )e"j!tdt
6.
Xs (!) =!!
n="!
x(nT )e"j!nT
where T is the sampling interval.
(a) Explain how the problems with computing the DTFT on a digitalcomputer can be overcome by choosing a uniformly spaced gripof N frequencies covering the range of ! from 0 to 2#/T andtruncating the summation in the expression above to just N datapoints; thus, determine the expression for the N -point DiscreteFourier Transform (DFT). [10%]
Xp =N"1!n=0
xne"j 2!N np, p = 0, . . . , N " 1
1
1. The Discrete Time Fourier Transform (DTFT) of a sampled sequence{xn}, n = 0,±1,±2, . . . ,±! is given by
X (!) =!!
n="!
xne"jn!T .
2.
X (!) =
!"
"!
x(t)e"j!tdt.
3.
xs(t) = x(t)!!
n="!
"(t " nT )
4.
Xs (!) =
!"
"!
xs(t)e"j!tdt
5.
Xs (!) =
!"
"!
x(t)!!
n="!
"(t " nT )e"j!tdt
6.
Xs (!) =!!
n="!
x(nT )e"j!nT
where T is the sampling interval.
(a) Explain how the problems with computing the DTFT on a digitalcomputer can be overcome by choosing a uniformly spaced gripof N frequencies covering the range of ! from 0 to 2#/T andtruncating the summation in the expression above to just N datapoints; thus, determine the expression for the N -point DiscreteFourier Transform (DFT). [10%]
Xp =N"1!n=0
xne"j 2!N np, p = 0, . . . , N " 1
1
1. The Discrete Time Fourier Transform (DTFT) of a sampled sequence{xn}, n = 0,±1,±2, . . . ,±! is given by
X (!) =!!
n="!
xne"jn!T .
2.
X (!) =
!"
"!
x(t)e"j!tdt.
3.
xs(t) = x(t)!!
n="!
"(t " nT )
4.
Xs (!) =
!"
"!
xs(t)e"j!tdt
5.
Xs (!) =
!"
"!
x(t)!!
n="!
"(t " nT )e"j!tdt
6.
Xs (!) =!!
n="!
x(nT )e"j!nT
where T is the sampling interval.
(a) Explain how the problems with computing the DTFT on a digitalcomputer can be overcome by choosing a uniformly spaced gripof N frequencies covering the range of ! from 0 to 2#/T andtruncating the summation in the expression above to just N datapoints; thus, determine the expression for the N -point DiscreteFourier Transform (DFT). [10%]
Xp =N"1!n=0
xne"j 2!N np, p = 0, . . . , N " 1
1
1. The Discrete Time Fourier Transform (DTFT) of a sampled sequence{xn}, n = 0,±1,±2, . . . ,±! is given by
X (!) =!!
n="!
xne"jn!T .
2.
X (!) =
!"
"!
x(t)e"j!tdt.
3.
xs(t) = x(t)!!
n="!
"(t " nT )
4.
Xs (!) =
!"
"!
xs(t)e"j!tdt
5.
Xs (!) =
!"
"!
x(t)!!
n="!
"(t " nT )e"j!tdt
6.
Xs (!) =!!
n="!
x(nT )e"j!nT
where T is the sampling interval.
(a) Explain how the problems with computing the DTFT on a digitalcomputer can be overcome by choosing a uniformly spaced gripof N frequencies covering the range of ! from 0 to 2#/T andtruncating the summation in the expression above to just N datapoints; thus, determine the expression for the N -point DiscreteFourier Transform (DFT). [10%]
Xp =N"1!n=0
xne"j 2!N np, p = 0, . . . , N " 1
1
1. The Discrete Time Fourier Transform (DTFT) of a sampled sequence{xn}, n = 0,±1,±2, . . . ,±! is given by
X (!) =!!
n="!
xne"jn!T .
2.
X (!) =
!"
"!
x(t)e"j!tdt.
3.
xs(t) = x(t)!!
n="!
"(t " nT )
4.
Xs (!) =
!"
"!
xs(t)e"j!tdt
5.
Xs (!) =
!"
"!
x(t)!!
n="!
"(t " nT )e"j!tdt
6.
Xs (!) =!!
n="!
x(nT )e"j!nT
where T is the sampling interval.
(a) Explain how the problems with computing the DTFT on a digitalcomputer can be overcome by choosing a uniformly spaced gripof N frequencies covering the range of ! from 0 to 2#/T andtruncating the summation in the expression above to just N datapoints; thus, determine the expression for the N -point DiscreteFourier Transform (DFT). [10%]
Xp =N"1!n=0
xne"j 2!N np, p = 0, . . . , N " 1
1
1. The Discrete Time Fourier Transform (DTFT) of a sampled sequence{xn}, n = 0,±1,±2, . . . ,±! is given by
X (!) =!!
n="!
xne"jn!T .
2.
X (!) =
!"
"!
x(t)e"j!tdt.
3.
xs(t) = x(t)!!
n="!
"(t " nT )
4.
Xs (!) =
!"
"!
xs(t)e"j!tdt
5.
Xs (!) =
!"
"!
x(t)!!
n="!
"(t " nT )e"j!tdt
6.
Xs (!) =!!
n="!
x(nT )e"j!nT
where T is the sampling interval.
(a) Explain how the problems with computing the DTFT on a digitalcomputer can be overcome by choosing a uniformly spaced gripof N frequencies covering the range of ! from 0 to 2#/T andtruncating the summation in the expression above to just N datapoints; thus, determine the expression for the N -point DiscreteFourier Transform (DFT). [10%]
Xp =N"1!n=0
xne"j 2!N np, p = 0, . . . , N " 1
1
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31
Discrete Time Fourier Transform – Signal Samples
Note that this expression known as DTFT is a periodic function of the frequency usually written as
The signal sample values may be expressed in terms of DTFT by noting that the equation above has the form of Fourier series (as a function of ω) and hence the sampled signal can be obtained directly as
[You can show this for yourself by first noting that (*) is a complex Fourier series with coefficients however it is also covered in one of Part IB Examples Papers]
1. The Discrete Time Fourier Transform (DTFT) of a sampled sequence{xn}, n = 0,±1,±2, . . . ,±! is given by
X (!) =!!
n="!
xne"jn!T .
2.
X (!) =
!"
"!
x(t)e"j!tdt.
3.
xs(t) = x(t)!!
n="!
"(t " nT )
4.
Xs (!) =
!"
"!
xs(t)e"j!tdt
5.
Xs (!) =
!"
"!
x(t)!!
n="!
"(t " nT )e"j!tdt
6.
Xs (!) =!!
n="!
x(nT )e"j!nT
where T is the sampling interval.
(a) Explain how the problems with computing the DTFT on a digitalcomputer can be overcome by choosing a uniformly spaced gripof N frequencies covering the range of ! from 0 to 2#/T andtruncating the summation in the expression above to just N datapoints; thus, determine the expression for the N -point DiscreteFourier Transform (DFT). [10%]
Xp =N"1!n=0
xne"j 2!N np, p = 0, . . . , N " 1
1
1. The Discrete Time Fourier Transform (DTFT) of a sampled sequence{xn}, n = 0,±1,±2, . . . ,±! is given by
X (!) =!!
n="!
xne"jn!T .
2.
X (!) =
!"
"!
x(t)e"j!tdt.
3.
xs(t) = x(t)!!
n="!
"(t " nT )
4.
Xs (!) =
!"
"!
xs(t)e"j!tdt
5.
Xs (!) =
!"
"!
x(t)!!
n="!
"(t " nT )e"j!tdt
6.
Xs (!) =!!
n="!
x(nT )e"j!nT
7.
xn =1
2#
2""
0
X(!)e+jn!Td!T
where T is the sampling interval.
(a) Explain how the problems with computing the DTFT on a digitalcomputer can be overcome by choosing a uniformly spaced gripof N frequencies covering the range of ! from 0 to 2#/T andtruncating the summation in the expression above to just N datapoints; thus, determine the expression for the N -point DiscreteFourier Transform (DFT). [10%]
Xp =N"1!n=0
xne"j 2!N np, p = 0, . . . , N " 1
1
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32
Computing DTFT on Digital Computer
The DTFT
expresses the spectrum of a sampled signal in terms of the signal samples but is not computable on a digital computer for two reasons:
1. The frequency variable ω is continuous. 2. The summation involves an infinite number of
samples.
1. The Discrete Time Fourier Transform (DTFT) of a sampled sequence{xn}, n = 0,±1,±2, . . . ,±! is given by
X (!) =!!
n="!
xne"jn!T .
2.
X (!) =
!"
"!
x(t)e"j!tdt.
3.
xs(t) = x(t)!!
n="!
"(t " nT )
4.
Xs (!) =
!"
"!
xs(t)e"j!tdt
5.
Xs (!) =
!"
"!
x(t)!!
n="!
"(t " nT )e"j!tdt
6.
Xs (!) =!!
n="!
x(nT )e"j!nT
where T is the sampling interval.
(a) Explain how the problems with computing the DTFT on a digitalcomputer can be overcome by choosing a uniformly spaced gripof N frequencies covering the range of ! from 0 to 2#/T andtruncating the summation in the expression above to just N datapoints; thus, determine the expression for the N -point DiscreteFourier Transform (DFT). [10%]
Xp =N"1!n=0
xne"j 2!N np, p = 0, . . . , N " 1
1
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33
Overcoming problems with computing DTFT
The problems with computing DTFT on a digital computer can be overcome by:
Step 1. Evaluating the DTFT at a finite collection of discrete frequencies.
no undesirable consequences, any frequency of interest can always be included in the collection
Step 2. Performing the summation over a finite number of data points
does have consequences since signals are generally not of finite duration
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34
The Discrete Fourier Transform (DFT)
The discrete set of frequencies chosen is arbitrary. However, since the DTFT is periodic we generally choose a uniformly spaced grid of N frequencies covering the range ωT from 0 to 2π. If the summation is then truncated to just N data points we get the DFT
The inverse DFT can be used to obtain the sampled signal values from the DFT: multiply each side by and sum over p=0 to N-1
Orthogonality property of complex exponentials
is N if n=q and 0 otherwise
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp = X(2"
Np) =
N!1!n=0
xne!j 2!N np, p = 0, . . . , N ! 1
(c) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =128kHz. Determine the total number of complex multiplicationsrequired. [20%].
Answer. 30 kHz corresponds to p = 15 since 15128kHz64 = 30kHz
(a uniformly spaced grip of N = 64 frequencies covering the rangeof ! from 0 to 2"/T is considered, where fs = 1/T = 128kHz )and one obtains
X15 =63!
n=0
xne!j 2!64
n15,
There are 63 complex multiplications (for n = 0 we multiply by1).
(d) The DFT is typically implemented using a Fast Fourier Transform(FFT) algorithm. Assuming N is even, split the summation in thebasic DFT equation into two parts: one for even n and one forodd n, and then show that the DFT values Xp and Xp+N/2 maybe expressed as
Xp = Ap + W pBp
Xp+N/2 = Ap ! W pBp
Derive Ap, Bp and W in the above expression, and thus define theFFT “butterfly” structure [40%]
2
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp = X(2"
Np) =
N!1!n=0
xne!j 2!N np, p = 0, . . . , N ! 1
(c) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =128kHz. Determine the total number of complex multiplicationsrequired. [20%].
Answer. 30 kHz corresponds to p = 15 since 15128kHz64 = 30kHz
(a uniformly spaced grip of N = 64 frequencies covering the rangeof ! from 0 to 2"/T is considered, where fs = 1/T = 128kHz )and one obtains
X15 =63!
n=0
xne!j 2!64
n15,
There are 63 complex multiplications (for n = 0 we multiply by1).
(d) The DFT is typically implemented using a Fast Fourier Transform(FFT) algorithm. Assuming N is even, split the summation in thebasic DFT equation into two parts: one for even n and one forodd n, and then show that the DFT values Xp and Xp+N/2 maybe expressed as
Xp = Ap + W pBp
Xp+N/2 = Ap ! W pBp
Derive Ap, Bp and W in the above expression, and thus define theFFT “butterfly” structure [40%]
2
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp = X(2"
Np) =
N!1!n=0
xne!j 2!N np, p = 0, . . . , N ! 1
(c)
N!1!p=0
Xpej 2!
N pq =N!1!p=0
N!1!n=0
xne!j 2!
N npp ej 2!
N pq =N!1!n=0
xn
N!1!p=0
ej 2!
N (q!n)pp
(d) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =128kHz. Determine the total number of complex multiplicationsrequired. [20%].
Answer. 30 kHz corresponds to p = 15 since 15128kHz64 = 30kHz
(a uniformly spaced grip of N = 64 frequencies covering the rangeof ! from 0 to 2"/T is considered, where fs = 1/T = 128kHz )and one obtains
X15 =63!
n=0
xne!j 2!64
n15,
There are 63 complex multiplications (for n = 0 we multiply by1).
(e) The DFT is typically implemented using a Fast Fourier Transform(FFT) algorithm. Assuming N is even, split the summation in thebasic DFT equation into two parts: one for even n and one forodd n, and then show that the DFT values Xp and Xp+N/2 maybe expressed as
Xp = Ap + W pBp
2
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp = X(2"
Np) =
N!1!n=0
xne!j 2!N np, p = 0, . . . , N ! 1
(c)
N!1!p=0
Xpej 2!
N pq =N!1!p=0
N!1!n=0
xne!j 2!
N npp ej 2!
N pq =N!1!n=0
xn
N!1!p=0
ej 2!
N (q!n)pp
(d) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =128kHz. Determine the total number of complex multiplicationsrequired. [20%].
Answer. 30 kHz corresponds to p = 15 since 15128kHz64 = 30kHz
(a uniformly spaced grip of N = 64 frequencies covering the rangeof ! from 0 to 2"/T is considered, where fs = 1/T = 128kHz )and one obtains
X15 =63!
n=0
xne!j 2!64
n15,
There are 63 complex multiplications (for n = 0 we multiply by1).
(e) The DFT is typically implemented using a Fast Fourier Transform(FFT) algorithm. Assuming N is even, split the summation in thebasic DFT equation into two parts: one for even n and one forodd n, and then show that the DFT values Xp and Xp+N/2 maybe expressed as
Xp = Ap + W pBp
2
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp = X(2"
Np) =
N!1!n=0
xne!j 2!N np, p = 0, . . . , N ! 1
(c)
N!1!p=0
Xpej 2!
N pq =N!1!p=0
N!1!n=0
xne!j 2!
N npp ej 2!
N pq =N!1!n=0
xn
N!1!p=0
ej 2!
N (q!n)pp
(d) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =128kHz. Determine the total number of complex multiplicationsrequired. [20%].
Answer. 30 kHz corresponds to p = 15 since 15128kHz64 = 30kHz
(a uniformly spaced grip of N = 64 frequencies covering the rangeof ! from 0 to 2"/T is considered, where fs = 1/T = 128kHz )and one obtains
X15 =63!
n=0
xne!j 2!64
n15,
There are 63 complex multiplications (for n = 0 we multiply by1).
(e) The DFT is typically implemented using a Fast Fourier Transform(FFT) algorithm. Assuming N is even, split the summation in thebasic DFT equation into two parts: one for even n and one forodd n, and then show that the DFT values Xp and Xp+N/2 maybe expressed as
Xp = Ap + W pBp
2
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp = X(2"
Np) =
N!1!n=0
xne!j 2!N np, p = 0, . . . , N ! 1
(c)
N!1!p=0
Xpej 2!
N pq =N!1!p=0
N!1!n=0
xne!j 2!
N npp ej 2!
N pq =N!1!n=0
xn
N!1!p=0
ej 2!
N (q!n)pp
(d) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =128kHz. Determine the total number of complex multiplicationsrequired. [20%].
Answer. 30 kHz corresponds to p = 15 since 15128kHz64 = 30kHz
(a uniformly spaced grip of N = 64 frequencies covering the rangeof ! from 0 to 2"/T is considered, where fs = 1/T = 128kHz )and one obtains
X15 =63!
n=0
xne!j 2!64
n15,
There are 63 complex multiplications (for n = 0 we multiply by1).
(e) The DFT is typically implemented using a Fast Fourier Transform(FFT) algorithm. Assuming N is even, split the summation in thebasic DFT equation into two parts: one for even n and one forodd n, and then show that the DFT values Xp and Xp+N/2 maybe expressed as
Xp = Ap + W pBp
2
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35
The Discrete Fourier Transform Pair
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36
• is periodic, for each p
• is periodic, for each n
• for real data
[You should check that you can show these results from first principles]
Properties of the Discrete Fourier Transform (DFT)
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp = X(2"
Np) =
N!1!n=0
xne!j 2!N np, p = 0, . . . , N ! 1
(c)
N!1!p=0
Xpej 2!
N pq =N!1!p=0
N!1!n=0
xne!j 2!
N npp ej 2!
N pq =N!1!n=0
xn
N!1!p=0
ej 2!
N (q!n)pp
Xp+N = Xp
(d)
xn+N = xn
(e)
Xp = X"
!p = X"
N!p
(f) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =128kHz. Determine the total number of complex multiplicationsrequired. [20%].
Answer. 30 kHz corresponds to p = 15 since 15128kHz64 = 30kHz
(a uniformly spaced grip of N = 64 frequencies covering the rangeof ! from 0 to 2"/T is considered, where fs = 1/T = 128kHz )and one obtains
X15 =63!
n=0
xne!j 2!64
n15,
There are 63 complex multiplications (for n = 0 we multiply by1).
2
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp = X(2"
Np) =
N!1!n=0
xne!j 2!N np, p = 0, . . . , N ! 1
(c)
N!1!p=0
Xpej 2!
N pq =N!1!p=0
N!1!n=0
xne!j 2!
N npp ej 2!
N pq =N!1!n=0
xn
N!1!p=0
ej 2!
N (q!n)pp
Xp+N = Xp
(d)
xn+N = xn
(e)
Xp = X"
!p = X"
N!p
(f) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =128kHz. Determine the total number of complex multiplicationsrequired. [20%].
Answer. 30 kHz corresponds to p = 15 since 15128kHz64 = 30kHz
(a uniformly spaced grip of N = 64 frequencies covering the rangeof ! from 0 to 2"/T is considered, where fs = 1/T = 128kHz )and one obtains
X15 =63!
n=0
xne!j 2!64
n15,
There are 63 complex multiplications (for n = 0 we multiply by1).
2
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp = X(2"
Np) =
N!1!n=0
xne!j 2!N np, p = 0, . . . , N ! 1
(c)
N!1!p=0
Xpej 2!
N pq =N!1!p=0
N!1!n=0
xne!j 2!
N npp ej 2!
N pq =N!1!n=0
xn
N!1!p=0
ej 2!
N (q!n)pp
Xp+N = Xp
(d)
xn+N = xn
(e)
Xp = X"
!p = X"
N!p
(f) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =128kHz. Determine the total number of complex multiplicationsrequired. [20%].
Answer. 30 kHz corresponds to p = 15 since 15128kHz64 = 30kHz
(a uniformly spaced grip of N = 64 frequencies covering the rangeof ! from 0 to 2"/T is considered, where fs = 1/T = 128kHz )and one obtains
X15 =63!
n=0
xne!j 2!64
n15,
There are 63 complex multiplications (for n = 0 we multiply by1).
2
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp = X(2"
Np) =
N!1!n=0
xne!j 2!N np, p = 0, . . . , N ! 1
(c)
N!1!p=0
Xpej 2!
N pq =N!1!p=0
N!1!n=0
xne!j 2!
N npp ej 2!
N pq =N!1!n=0
xn
N!1!p=0
ej 2!
N (q!n)pp
Xp+N = Xp
(d)
xn+N = xn
(e)
Xp = X"
!p = X"
N!p
(f) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =128kHz. Determine the total number of complex multiplicationsrequired. [20%].
Answer. 30 kHz corresponds to p = 15 since 15128kHz64 = 30kHz
(a uniformly spaced grip of N = 64 frequencies covering the rangeof ! from 0 to 2"/T is considered, where fs = 1/T = 128kHz )and one obtains
X15 =63!
n=0
xne!j 2!64
n15,
There are 63 complex multiplications (for n = 0 we multiply by1).
2
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp = X(2"
Np) =
N!1!n=0
xne!j 2!N np, p = 0, . . . , N ! 1
(c)
N!1!p=0
Xpej 2!
N pq =N!1!p=0
N!1!n=0
xne!j 2!
N npp ej 2!
N pq =N!1!n=0
xn
N!1!p=0
ej 2!
N (q!n)pp
Xp+N = Xp
(d)
xn+N = xn
(e)
Xp = X"
!p = X"
N!p
(f) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =128kHz. Determine the total number of complex multiplicationsrequired. [20%].
Answer. 30 kHz corresponds to p = 15 since 15128kHz64 = 30kHz
(a uniformly spaced grip of N = 64 frequencies covering the rangeof ! from 0 to 2"/T is considered, where fs = 1/T = 128kHz )and one obtains
X15 =63!
n=0
xne!j 2!64
n15,
There are 63 complex multiplications (for n = 0 we multiply by1).
2
(b) Answer. There are two problems with calculating DTFT on a dig-ital computer: first, the frequency variable ! is continuous whichcan be overcome by evaluating the DTFT at a finite collectionof discrete frequencies without any undesirable consequences; sec-ond, the summation involves infinite number of samples which canbe overcome by performing the summation over a finite numberof data points, which does have does have consequences since sig-nals are generally not of finite duration. If we choose a uniformlyspaced grip of N frequencies covering the range of ! from 0 to2"/T and truncate the summation in the expression above to justN data points we get the DFT:
Xp = X(2"
Np) =
N!1!n=0
xne!j 2!N np, p = 0, . . . , N ! 1
(c)
N!1!p=0
Xpej 2!
N pq =N!1!p=0
N!1!n=0
xne!j 2!
N npp ej 2!
N pq =N!1!n=0
xn
N!1!p=0
ej 2!
N (q!n)pp
Xp+N = Xp
(d)
xn+N = xn
(e)
Xp = X"
!p = X"
N!p
(f) Assume you are interested in detecting the presence of a singlecontinuous-wave sinusoidal tone in a time-domain data sequence.Show how the 64-point DFT of {xn} can be computed at thediscrete frequency of 30kHz if the sampling rate of {xn} was fs =128kHz. Determine the total number of complex multiplicationsrequired. [20%].
Answer. 30 kHz corresponds to p = 15 since 15128kHz64 = 30kHz
(a uniformly spaced grip of N = 64 frequencies covering the rangeof ! from 0 to 2"/T is considered, where fs = 1/T = 128kHz )and one obtains
X15 =63!
n=0
xne!j 2!64
n15,
There are 63 complex multiplications (for n = 0 we multiply by1).
2
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DFT Interpolation
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Zero padding
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Padded sequence
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Zero-padding
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Zero-padding
just visualisation, not additional information!
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Circular Convolution
circular convolution
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Example of Circular Convolution
1
2 0
3
5 4
Circular convolution of x1={1,2,0} and x2={3,5,4} clock-wise anticlock-wise
1
2 0
3
5 4
1
2 0
5
4 3
1
2 0
4
3 5 0 spins 1 spin 2 spins
y(0)=1×3+2×4+0×5 y(1)=1×5+2×3+0×4 y(2)=1×4+2×5+0×3
folded sequence
x1(n)x2(0-n)|mod3 x1(n)x2(1-n)|mod3 x1(n)x2(2-n)|mod3
…
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Example of Circular Convolution
clock-wise anticlock-wise
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Example of Circular Convolution
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Standard Convolution using Circular Convolution
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Example of Circular Convolution
1
2 0 3
5 0
clock-wise anticlock-wise 1
2 3
5 0
0 spins
y(0)=1×3+2×0+0×4+0×5
folded sequence
x1(n)x2(0-n)|mod3 x1(n)x2(1-n)|mod3 x1(n)x2(2-n)|mod3
…
0 4
4
0
1
2 0 5
4 3 1 spin
0
0
1
2 0 4
0 5 2 spins
3
0
0 y(0)=1×5+2×3+0×0+0×4 y(0)=1×4+2×5+0×3+0×0
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Standard Convolution using Circular Convolution
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Proof of Validity
Circular convolution of the padded sequence corresponds to the standard convolution
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Linear Filtering using the DFT
FIR filter:
DFT and then IDFT can be used to compute standard convolution product and thus to perform linear filtering.
Frequency domain equivalent:
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Summary So Far
• Fourier analysis for periodic functions focuses on the study of Fourier series
• The Fourier Transform (FT) is a way of transforming a continuous signal into the frequency domain
• The Discrete Time Fourier Transform (DTFT) is a Fourier Transform of a sampled signal
• The Discrete Fourier Transform (DFT) is a discrete numerical equivalent using sums instead of integrals that can be computed on a digital computer
• As one of the applications DFT and then Inverse DFT (IDFT) can be used to compute standard convolution product and thus to perform linear filtering
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Fast Fourier Transform (FFT)
Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205
Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet,
Dr. Malcolm Macleod and Prof. Peter Rayner
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Computations for Evaluating the DFT
• Discrete Fourier Transform and its Inverse can be computed on a digital computer
• What are computational requirements?
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Operation Count
How do we evaluate computational complexity? count the number of real multiplications and
additions by ¨real¨ we mean either fixed-point or floating point
operations depending on the specific hardware
subtraction is considered equivalent to additions divisions are counted separately
Other operations such as loading from memory, storing in memory, loop counting, indexing, etc are not counted (depends on implementation/architecture) – present overhead
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Operation Count – Modern DSP
Modern DSP: a real multiplication and addition is a single machine
cycle y=ax+b called MAC (multiply/accumulate)
maximum number of multiplications and additions must be counted, not their sum
traditional claim ¨multiplication is much more time- consuming than addition¨ becomes obsolete
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Discrete Fourier Transform as a Vector Operation
Again, we have a sequence of sampled values x and transformed values X , given by
n
p
Let us denote:
Then:
x
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x
X 0 0x 0 0 0 0…X 1 1x 0 1 2 N-1 …
…… …… … … …
…X N-1 0 N-1 2(N-1) (N-1) 2 x N-1
=
This Equation can be rewritten in matrix form:
as follows:
Matrix Interpretation of the DFT
N x N matrix
complex numbers
can be computed off-line and stored
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DFT Evaluation – Operation Count
Multiplication of a complex N x N matrix by a complex N-dimensional vector.
2 Total: 4(N-1) real ¨x¨s & 4(N-0.5)(N-1) real ¨+¨s
Complex ¨x¨: 4 real ¨x¨s and 2 real ¨+¨s; complex ¨+¨ : 2 real ¨+¨s
2 First row and first column are 1 however
save 2N-1 ¨x¨s, (N-1) complex ¨x¨s & N(N-1) ¨+¨s
2
No attempt to save operations N complex multiplications (¨x¨s)
N(N-1) complex additions (¨+¨s)
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DFT Computational Complexity
Thus, for the input sequence of length N the number of arithmetic operations in direct computation of DFT is proportional to N . 2
For N=1000, about a million operations are needed!
In 1960s such a number was considered prohibitive in most applications.
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Discovery of the Fast Fourier Transform (FFT)
When in 1965 Cooley and Tukey ¨first¨ announced discovery of Fast Fourier Transform (FFT) in 1965 it revolutionised Digital Signal Processing.
They were actually 150 years late – the principle of the FFT was later discovered in obscure section of one of Gauss’ (as in Gaussian) own notebooks in 1806.
Their paper is the most cited mathematical paper ever written.
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The FFT is a highly elegant and efficient algorithm, which is still one of the most used algorithms in speech processing, communications, frequency estimation, etc – one of the most highly developed area of DSP.
There are many different types and variations.
They are just computational schemes for computing the DFT – not new transforms!
Here we consider the most basic radix-2 algorithm which requires N to be a power of 2.
The Fast Fourier Transform (FFT)
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Lets take the basic DFT equation:
Now, split the summation into two parts: one for even n and one for odd n:
Take outside
FFT Derivation 1
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Thus, we have:
FFT Derivation 2
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Now, take the same equation again where we split the summation into two parts: one for even n and one for odd n:
And evaluate at frequencies p+N/2
FFT Derivation 3
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FFT Derivation 4
simplified
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=
=
FFT Derivation 5
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FFT Derivation - Butterfly
Or, in more compact form:
(‘Butterfly’)
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FFT Derivation - Redundancy
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FFT Derivation - Computational Load
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Flow Diagram for a N=8 DFT
Input: Output:
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Flow Diagram for a N=8 DFT - Decomposition
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Flow Diagram for a N=8 DFT – Decomposition 2
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Bit-reversal
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FFT Derivation Summary
• The FFT derivation relies on redundancy in the calculation of the basic DFT
• A recursive algorithm is derived that repeatedly rearranges the problem into two simpler problems of half the size
• Hence the basic algorithm operates on signals of length a power of 2, i.e.
(for some integer M)
• At the bottom of the tree we have the classic FFT `butterfly’ structure
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Computational Load of full FFT algorithm
The type of FFT we have considered, where N = 2M, is called a radix-2 FFT.
It has M = log2 N stages, each using N / 2 butterflies
Complex multiplication requires 4 real multiplications and 2 real additions
Complex addition/subtraction requires 2 real additions
Thus, butterfly requires 10 real operations.
Hence the radix-2 N-point FFT requires 10( N / 2 )log2 N real operations compared to about 8N2 real operations for the DFT.
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Computational Load of full FFT algorithm
Direct DFT
FFT
The radix-2 N-point FFT requires 10( N / 2 )log2 N real operations compared to about 8N2 real operations for the DFT.
This is a huge speed-up in typical applications, where N is 128 – 4096:
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Input Output
Further advantage of the FFT algorithm
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The Inverse FFT (IFFT)
The IDFT is different from the DFT:
• it uses positive power of instead of negative ones • There is an additional division of each output value by N
Any FFT algorithm can be modified to perform the IDFT by • using positive powers instead of negatives • Multiplying each component of the output by 1 / N
Hence the algorithm is the same but computational load increases due to N extra multiplications
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The form of FFT we have described is called “decimation in time”; there is a form called “decimation in frequency” (but it has no advantages).
The "radix 2" FFT must have length N a power of 2. Slightly more efficient is the "radix 4" FFT, in which 2-input 2-output butterflies are replaced by 4-input 4-output units. The transform length must then be a power of 4 (more restrictive).
A completely different type of algorithm, the Winograd Fourier Transform Algorithm (WFTA), can be used for FFT lengths equal to the product of a number of mutually prime factors (e.g. 9*7*5 = 315 or 5*16 = 80). The WFTA uses fewer multipliers, but more adders, than a similar-length FFT.
Efficient algorithms exist for FFTing real (not complex) data at about 60% the effort of the same-sized complex-data FFT.
The Discrete Cosine and Sine Transforms (DCT and DST) are similar real-signal algorithms used in image coding.
Many types of FFT
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There FFT is surely the most widely used signal processing algorithm of all. It is the basic building block for a large percentage of algorithms in current usage
Specific examples include: • Spectrum analysis – used for analysing and detecting signals • Coding – audio and speech signals are often coded in the frequency domain using FFT variants (MP3, …) • Another recent application is in a modulation scheme called OFDM, which is used for digital TV broadcasting (DVB) and digital radio (audio) broadcasting (DAB). • Background noise reduction for mobile telephony, speech and audio signals is often implemented in the frequency domain using FFTs
Applications of the FFT
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Practical Spectral Analysis
• Say, we have a symphony recording over 40 minutes long – about 2500 seconds
• Compact-disk recordings are sampled at 44.1 kHz and are in stereo, but say we combine both channels into a single one by summing them at each time point
• Are we to compute the DFT of a sequence one hundred million samples long?
a) unrealistic – speed and storage requirements are too high
b) useless – we will get a very much noiselike wide-band spectrum covering the range from 20 Hz to 20 kHz at high resolution including all notes of all instruments with their harmonics
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• We actually want a sequence of short DFTs, each showing the spectrum of a signal of relatively short interval
• If violin plays note E during this interval – spectrum would exhibit energies at not the frequency of note E (329 Hz) and characteristic harmonics of the violin
• If the interval is short enough – we may be able to track not-to-note changes
• If the frequency resolution is high enough – we will be able to identify musical instruments playing
Remember: our – ear – excellent spectrum analyser?
Short-Time Spectral Analysis
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• If we take a time interval of about 93 milliseconds (signal of length 4096) the frequency resolution will be 11 Hz (adequate)
• The number of distinct intervals in the symphony is 40 x 60/0.093 - about 26,000
• We may choose to be conservative and overlap the intervals (say 50%)
• In total, need to compute 52,000 FFTs of length 4096
the entire computation will be done in considerably less time than the music itself
Short-time spectral analysis also use windowing (more on this later)
Short-Time Spectral Analysis – Computational Load
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Case Study: Spectral analysis of a Musical Signal
Extract a short segment
Looks almost periodic over short time interval
Sample rate is 10.025 kHz (T=1/10,025 s)
Load this into Matlab as a vector x
Take an FFT, N=512
X=fft(x(1:512));
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Note Conjugate symmetry as data are real: Symmetric
Symmetric Anti-Symmetric
Case Study: Spectral analysis of a Musical Signal, FFT
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Case Study: Spectral analysis of a Musical Signal - Frequencies
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The Effect of data length N
N=32
N=128
N=1024
FFT
Low resolution
High resolution
FFT
FFT
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The DFT approximation to the DTFT
DTFT at frequency : DFT:
• Ideally the DFT should be a `good’ approximation to the DTFT
• Intuitively the approximation gets better as the number of data points N increases
• This is illustrated in the previous slide – resolution gets better as N increases (more, narrower, peaks in spectrum).
• How to evaluate this analytically? View the truncation in the summation as a multiplication by a
rectangle window function
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Analysis
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Pre-multiplying by rectangular window
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Spectrum of the windowed signal
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Fourier Transform of the Rectangular Window
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N=32 Central `Lobe’
Sidelobes
Rectangular Window Spectrum
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N=4
N=8
N=16
N=32
Lobe width inversely Proportional to N
Rectangular Window Spectrum – Lobe Width
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Now, imagine what happens when the sum of two frequency components is DFT-ed:
The DTFT is given by a train of delta functions:
Hence the windowed spectrum is just the convolution of the window spectrum with the delta functions:
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Both components separately
Both components Together
ωΤ
DFT for the data
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Brief summary
• The rectangular window introduces broadening of any frequency components (`smearing’) and sidelobes that may overlap with other frequency components (`leakage’).
• The effect improves as N increases • However, the rectangle window has poor properties and better choices
of wn can lead to better spectral properties (less leakage, in particular) – i.e. instead of just truncating the summation, we can pre-multiply by a suitable window function wn that has better frequency domain properties.
• More on window design in the filter design section of the course – see later
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Summary so far … • The Fourier Transform (FT) is a way of transforming
a continuous signal into the frequency domain
• The Discrete Time Fourier Transform (DTFT) is a Fourier Transform of a sampled signal
• The Discrete Fourier Transform (DFT) is a discrete numerical equivalent using sums instead of integrals that can be computed on a digital computer
• The FFT is a highly elegant and efficient algorithm, which is still one of the most used algorithms in speech processing, communications, frequency estimation, etc – one of the most highly developed area of DSP
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Thank you!