Fourier Analysis & Discrete-time Systems - …numbat.ucsd.edu/~bob/signals2015Winter/Signals_&_Systems_2015... · Discrete-time signals & systems ... Direct solution ... Moving from

Fourier Analysis & Discrete-time Systems

Road map – looking backwards Continuous-time signals & systems

Ordinary differential equations Integrating factors, variation of parameters

State-space and vector ODEs

Zero-input response + zero-state response

Impulse response, step response

Convolution

Laplace transform methods

Transfer functions, poles, zeros

Partial fraction expansion

Frequency response, sinusoidal steady-state, Bode diagrams Filtering; lowpass, highpass, etc

Signals & systems; inputs, states, outputs, responses MAE143A Signals & Systems Winter 2015 1


Road map – looking forwards Sampling and discrete-time signals

Periodic signals & Fourier series Complex Fourier series

Line spectrum, Parseval

Frequency response

Fourier transform

bandlimited, convolution

Periodically sampled signals

Fourier transform of sampled signals

ADC, lowpass filtering

sampling and reconstruction

Aliasing

Discrete Fourier Transform (DFT/FFT)

MAE143A Signals & Systems Winter 2015 2

Chaparro 4.3 4.4 4.9 5.2 5.7 7.2 7.4 7.2-7.3 10.2-10.4


Road map – looking forwards Discrete-time signals & systems

Ordinary difference equations Direct solution

State-space and vector ODEs

Zero-input response + zero-state response

Impulse response, step response

Convolution

Z-transform methods

Transfer functions, poles, zeros

Partial fraction expansion

Frequency response, sinusoidal steady-state, Bode diagrams Filtering; lowpass, highpass, etc

Signals & systems; inputs, states, outputs, responses MAE143A Signals & Systems Winter 2015 3

Moving from continuous- to discrete-time

Sampling ideas Uniform, periodic sampling rate, e.g. CDs at 44.1KHz

First we will need to consider periodic signals in order to appreciate how to interpret discrete-time signals as representative of continuous-time

Fourier series and transforms

Discrete-time Fourier transform (DTFT)

Discrete Fourier transform (DFT) & the fast Fourier transform (FFT)

Aliasing and signal reconstruction

Discrete-time signals then discrete-time systems Matlab tools and digital signal processing

Nothing on digital (finite bit-length) signals


Fourier series for periodic signals (Chap 3)

Periodic signal For all times Period here is 6 sec

Smallest possible repetition time

The signal is real

Fourier series representation Sometimes called the complex Fourier series representation


x(t+ T ) = x(t), 8t 2 (�1,1)

t 2 (�1,1)

x(t) =1X

k=�1cke

jk�0t, t 2 (�1,1), ⇥0 =2�

T

Fourier series for periodic signals

Signal x(t) is periodic with fundamental period T

Fourier coefficients are complex numbers Since x(t) is real

The trigonometric Fourier series


x(t) =1X

k=�1cke

jk�0t, t 2 (�1,1), ⇥0 =2�

T

ck =1

T

Z T

0x(t)e�jk!0t

dt, k = . . . ,�2,�1, 0, 1, 2, . . .

c�k = c̄k

x(t) = c0 + 2

1X

k=1

|ck| cos(k!0t+ \ck), t 2 (�1,1)

= a0 +

1X

k=1

[ak cos(k!0t) + bk sin(k!0t)]

The complex exponential Fourier series


x(t) =1X

k=�1cke

jk�0t, t 2 (�1,1), ⇥0 =2�

T

ck =1

T

Z T

0x(t)e�jk!0t

dt, k = . . . ,�2,�1, 0, 1, 2, . . .

The Phantom is not afraid of complex numbers

He always uses the complex Fourier series representation of periodic signals

So do we!

Computing Fourier coefficients


1T

Z t+T

tejk 2⇡

T ⌧ej` 2⇡T ⌧ d⌧ =

1T

Z t+T

tej(k+`) 2⇡

T ⌧ d⌧

=1T

T

j(k + `)2⇡ej(k+`) 2⇡

T ⌧

��t+T

t

=1

j(k + `)2⇡

hej(k+`) 2⇡

T (t+T ) � ej(k+`) 2⇡T t

i

=ej(k+`) 2⇡

T t

j(k + `)2⇡

hej(k+`) 2⇡

T T � 1i

=

(0, k 6= �`

1, k = �`

k 6= �` : ej(k+`) 2⇡T T � 1 = 1� 1 = 0

k = �` :

ej(k+`) 2⇡T

t

j(k+`)2⇡

hej(k+`) 2⇡

T T � 1i

= ej0

j(k+`)2⇡

h1 + {j(k + `)2⇡} + {j(k+`)2⇡}2

2! + {j(k+`)2⇡}3

3! + · · ·� 1i

= 1j(k+`)2⇡

hj(k + `)2⇡ + {j(k+`)2⇡}2

2! + {j(k+`)2⇡}3

3! + . . .i

= 1 + {j(k+`)2⇡}2! + {j(k+`)2⇡}2

3! + · · · = 1

Fourier series example – square wave

T=1, ω0=2π

Compute coefficients


x(t) =

(0, t 2 (0, 0.5]

1, t 2 (0.5, 1]

c0 =1

1

Z 1

0x(t) dt =

Z 1

0.5dt = 0.5

ck =

Z 1

0x(t)e

�jk2⇡tdt =

Z 1

0.5e

�jk2⇡tdt =

1

�jk2⇡

e

�jk2⇡t

��1

0.5

=

1

�jk2⇡

�e

�jk2⇡ � e

�jk⇡�=

(jk⇡ , k odd

0, k even

k = . . . ,�2,�1, 1, 2, . . .

Fourier series example –square wave


x(t) = 0.5 +1

�

1X

k=1

j

2k � 1ej(2k�1)2�t � j

2k � 1e�j(2k�1)2�t

= 0.5� 2

�

1X

k=1

1

2k � 1sin [2�(2k � 1)t]

Square wave matlab code t4=[-1.3:.001:2.3]ʼ’;

x4=round(mod(t4,1));

axis([-2 3 -0.3 1.3]);

a0=0.5*ones(size(t4));

a1=-2/pi*sin(2*pi*t4);

a3=-2/3/pi*sin(6*pi*t4);

a5=-2/5/pi*sin(10*pi*t4);

plot(t4,a0,ʼ’gʼ’,t4,a0+a1,ʼ’rʼ’,t4,a0+a1+a3,ʼ’mʼ’,t4,a0+a1+a3+a5,ʼ’kʼ’,t4,x4,ʼ’bʼ’,ʼ’Linewidthʼ’,2)

legend(ʻ‘one termʼ’,ʼ’two termsʼ’,ʼ’three termsʼ’,ʼ’four termsʼ’)

xlabel(ʻ‘Time (sec)ʼ’)

ylabel(ʻ‘Signal amplitude (units)ʼ’)

title(ʻ‘Periodic square wave signalʼ’)


MAE143A Signals & Systems Winter 2015 12 MAE143A Signals & Systems Winter 2005

12

Line spectrum of the square wave signal

Plot the component amplitudes versus frequency Fourier tells us this is mostly an empty plot


Parseval’s formula

The power of a signal can be measured in the time or in the frequency domain

Power = energy per period/period length


P =1

T

Z T

0x

2(t) dt

=1

T

Z T

0

" 1X

k=�1cke

jk 2⇡T t

#2

dt

=1X

k=�1

1X

`=�1

1

T

Z T

0ckc`e

jk 2⇡T t

e

j` 2⇡T t

dt

=1X

k=�1|ck|2

Parseval’s formula and signal power

Square wave example Time domain Frequency domain


P =1

T

Z T

0x(t)2 dt =

1X

k=�1|ck|2

P =1

T

Z T

0x(t)2 dt =

Z 1

0.51 dt = 0.5W

1X

k=�1|ck|2 = 0.25 +

2

�2

1X

k=1

1

(2k � 1)2= 0.5W

Summary of Fourier series

Write a periodic signal as an infinite sum of complex exponentials at harmonic frequencies

The line spectrum depicts the frequency content Parseval tells us this captures all the energy properly

A linear system with periodic input (not just sinusoidal) will have a steady-state response which is the sum of the individual s-s-s responses

The contribution from each component is given by the systems frequency response


Frequency response example


System 1: G1(s) =20

s+ 20

System 2: G2(s) =23000

s+ 20s+ 23000


System 1: G1(s) =20

s+ 20

System 2: G2(s) =23000

s+ 20s+ 23000

uk =1X

k=�1cke

jk 2⇡T t

yi,k =1X

k=�1di,ke

jk 2⇡T t

di,k = Gi

✓jk

2⇡

T

◆ck

The Fourier transform (Chaparro Chap 5)

Generalization of the Fourier series to non-periodic function – only one single, very long period

X(ω) is like the line spectrum with zero spacing between harmonic elements

Still represents the signal as a sum of exponentials Requires absolute integrability of x(t)- finite energy

Inverse Fourier transform


T ! 1

X(�) =

Z 1

�1x(t)e�j�t dt

x(t) =1

2⇡

Z 1

�1X(!)ej!t

d!

The Fourier transform - example

Two-sided decaying exponential


X(!) =

Z 1

�1e�

|t|5 e�j!t dt

=

Z 0

�1e

t5�j!t dt+

Z 1

0e�

t5�j!t dt

=�1

j! � 15

+1

j! + 15

|X(!)| =25

!2 + 125

x(t) = e

� |t|5, t 2 (�1,1)

The Fourier and Laplace transforms

For a one-sided signal x(t)with Laplace transform

Note that this is a signal property It mimics the Laplace transform and frequency response relationship for causal systems

The interest in Fourier transforms is in the infinite-time response to periodic inputs

Special Fourier transforms


X(!) = X (s)|s=j!

X (s)

Y (!) = H(j!)U(!), H(!) = F(h(t))

F [�(t)] = 1, F [cos!0t] = ⇡[�(! � !0) + �(! + !0)]

F [1] = 2⇡�(!), F [sin!0t] = j⇡[��(! � !0) + �(! + !0)]

Bandlimited signals

Signals, x(t), whose Fourier transform, X(ω), is zero for frequencies outside a value, the bandwidth, B, i.e.

is referred to as bandlimited

A periodic signal with line spectrum zero outside the bandwidth B is also referred to as bandlimited

Bandlimited implies a level of smoothness (as we shall see in the homework)

Fourier series: finite power, periodic

Fourier transform: finite energy, one-shot MAE143A Signals & Systems Winter 2015 22

X(!) = 0 for {! > B} [ {! < �B}

Convolution and the Fourier transform

The same as for the Laplace transform


x(t) ⇤ y(t) =Z 1

�1x(t� ⌧)y(⌧) d⌧ = F�1 [X(!)Y (!)]

F(x ⇥ y) =

Z 1

�1

Z 1

�1x(t� ⇥)y(⇥) d⇥

�e�j⇤t dt

=

Z 1

�1

Z 1

�1x(t� ⇥)e�j⇤(t�⇥)y(⇥)e�j⇤⇥ d⇥ dt

=

Z 1

�1x(t� ⇥)e�j⇤(t�⇥) dt

Z 1

�1y(⇥)e�j⇤⇥ d⇥

=

Z 1

�1x(�)e�j⇤� d�

Z 1

�1y(⇥)e�j⇤⇥ d⇥

= X(⇤)Y (⇤)

� = t� ⌧d� = �d⌧,⌧ = �1 ! � = 1⌧ = 1 ! � = �1

Convolution and the Fourier transform II

Using the symmetry between and


X(�) =

Z 1

�1x(t)e�j�t dt x(t) =

1

2⇡

Z 1

�1X(!)ej!t

d!

F F�1

F [x(t)y(t)] =1

2⇡X(!) ⇤ Y (!) =

1

2⇡

Z 1

�1X(! � �)Y (�) d�

F [x(t)y(t)] =

Z 1

t,�1x(t)y(t)e�j!t

dt

=

Z 1

t,�1

1

2⇡

Z 1

⇢,�1X(⇢)ej⇢t d⇢

� 1

2⇡

Z 1

�,�1Y (�)ej�t d�

�e

�j!tdt

=1

4⇡2

Z 1

⇢,�1X(⇢)

⇢Z 1

�,�1Y (�)

Z 1

t,�1e

�j!te

j⇢te

j�tdt

�d�

�d⇢

=1

4⇡2

Z 1

⇢,�1X(⇢)

⇢Z 1

�,�1Y (�)⇥ 2⇡�(⇢+ � � !) d�

�d⇢

=1

2⇡

Z 1

⇢,�1X(⇢)Y (! � ⇢) d⇢ =

1

2⇡X(!) ⇤ Y (!)

Convolution and the Fourier transform III

The remarkable symmetry between forward and inverse Fourier transforms means that convolution in one domain (time or frequency) corresponds to multiplication in the other domain

In digital signal processing, we often use a technique known as windowing

These involve time-domain multiplication Check out matlab doc on ‘window’

Hanning, Hamming, Bartlett, Blackman, Flattop, Chebyshev, Kaiser, Gaussian, Nuttall, Rectangular, Parzen, Taylor, Triangular, Tukey, …


Sampled signals – discrete time

For computer use we need to sample signals Most commonly this is periodic sampling every T seconds we take a sample of the continuous signal This yields a sequence

A continuous-time signal is a function of time t A discrete-time signal is a sequence indexed by t

T is the sampling period (sec)

1/T is the sample rate (Hz)

Often we also quantize the samples to yield a digital signal of a fixed number of bits representation

This is a subject for the next course


xn = x(nT )

Analog-to-digital conversion

Signal conditioning Filtering (anti-aliasing), offset, ranging

Track and hold circuit Enabled by the rising edge trigger Holds the signal at the latched value

Analog-to-digital converter Converts held signal to 8-bit, 16-bit, 24-bit digital

Computer – coordinates sampling, storage, 1/T sec MAE143A Signals & Systems Winter 2015 27

Electronic Signal

Conditioner Real-world signal

Track & hold Circuit ADC µC

Ideal sampling of signals

Continuous-time signal x(t) Continuous values of x, t

Discrete-time signal xn

Continuous x, integer n A sequence

Sampled-data signal xs(t) Continuous x, t Modeled by a pulse train


t

x(t)

t

xs(t)

0 T 2T 3T 4T 5T 6T 7T -T -2T

xn

0 1 2 3 4 5 6 7 -1 -2 n

xs(t) =1X

n=�1xn�(t� nT ) = x(t)⇥

1X

n=�1�(t� nT )

Fourier transform of a sampled signal

Fourier transform of a product is the convolution of the Fourier transforms

What is the Fourier series of periodic pulse train p(t)?


xs(t) =1X

n=�1xn�(t� nT ) = x(t)⇥

1X

n=�1�(t� nT ) = x(t)p(t)

p(t) =1X

k=�1cke

jkwst

ck =1T

Z T/2

�T/2p(t)e�jk!st dt =

1T

p(t) =1X

k=�1

1T

ejk!st

Fourier transform of a sampled signal

Periodic pulse-train

Sampled signal

Fourier transform is immediate Repeated addition of the transform of x(t) in the frequency domain

Relies on the property


!s =2⇡

T

F⇥x(t)ej!0t

⇤= X(! � !0)

p(t) =1X

k=�1

1T

ejk!st

x(t)p(t) =1X

k=�1

1T

x(t)ejk!st

Xs(!) =1X

k=�1

1T

X(! � k!s)

Fourier transform of a sampled signal II

Fourier transform of a product is the convolution of the Fourier transforms

The spectrum of a perfectly sampled signal xs(t) is a repeated and scaled version of X(ω)


xs(t) =1X

n=�1xn�(t� nT ) = x(t)⇥

1X

n=�1�(t� nT ) = x(t)p(t)

X(ω)

ω

Xs(ω)

ωωs 2ωs-ωs 0

p(t) =1X

n=�1�(t� nT ) , P (!) =

2⇡

T

1X

k=�1�(! � k!s), !s =

2⇡

T

Xs(!) =1

2⇡

Z 1

�1X(! � �)P (�) d� =

1

T

1X

k=�1X(! � k!s)

Fourier transform of a sampled signal III

The spectrum of a perfectly sampled signal xs(t) is a repeated and scaled version of X(ω)

Bandlimited signal bandwidth B<ωs/2 Signal with bandwidth B>ωs/2

Expect trouble in reconstruction of x(t) from xs(t)


X(ω)

ω

Xs(ω)

ωωs 2ωs-ωs 0

X(ω)

ω

Xs(ω)

ωωs 2ωs-ωs 0

Reconstructing sampled bandlimited signals

Reconstruction of the original signal from samples

We can reconstruct using an ideal lowpass filter This only works if

is called the Nyquist sampling frequency


Xs(ω)

ωωs 2ωs-ωs 0B -B

H(!)x(t)xs(t)

Lowpass Filter Bandwidth B

!s � 2B

A signal of bandwidth B can be reconstructed completely and exactly from the sampled signal if the sampling frequency ωs is greater than 2B

Nyquist’s Sampling Theorem

!s = 2B

Ideal lowpass filter I

Ideal lowpass filter


H(!)x(t)xs(t)


-B B

h(t) =1

2⇡

Z 1

�1H(!)e�j!t d! =

A

2⇡

Z B

�Be�j!t d!

=A

2⇡

1

�jte�j!t

��B

�B

=A

2⇡

2

t

1

2j

⇥ejBt � e�jBt

⇤

=AB

⇡

sinBt

Bt

4=

AB

⇡sincBt

H(!)

!

A

Ideal lowpass filter II

Ideal lowpass filter impulse response – sinc function

MAE143A Signals & Systems Winter 2015 35 −4 −3 −2 −1 0 1 2 3 4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)

Sign

al (u

nits

)

sinc function plot: sinc(x)=sin(! x)/! x

H(!) =

(A, ! 2 [�B,B]

0, else, h(t) =

AB

⇡sincBt

Non-causal non-timelimited impulse response Problems in realization c.f. matlab: butter, cheby, ellip, etc

Shannon’s interpolation formula

Convolution formula


H(!)x(t)xs(t)


x(t) = h(t) ⇤ xs(t)

=

Z 1

�1

⇢BT

⇡

sinc

B

⇡

(t� ⌧)

��( 1X

n=�1x(nT )�(⌧ � nT )

)d⌧

=BT

⇡

1X

n=�1x(nT )

⇢Z 1

�1sinc

B

⇡

(t� ⌧)

��(⌧ � nT ) d⌧

�

=BT

⇡

1X

n=�1x(nT ) sinc

B

⇡

(t� nT )

�

Sampling and reconstruction

If the continuous-time signal is bandlimited to B and we sample faster than the Nyquist frequency

Then we can exactly reconstruct the original signal from the sample sequence

Conversely: if we sample at or more slowly than the Nyquist frequency then we cannot reconstruct the original signal from the samples alone

We might have other information that helps the reconstruction. Without such extra information we suffer from Aliasing – some frequencies masquerading as others


Aliasing

Sampling rate is less than the Nyquist rate Spectra overlap in sampled signal Impossible to disentangle

Signals from above ωs/2 masquerade as signals of a lower frequency

This is what happens to wagon wheels in western movies

fs=30Hz but the passing rate of the spokes is greater than this

Dopplegänger and impersonation

Alias: a false or assumed identity

Samuel Langhorne Clemens alias Mark Twain

5.5KHz sampled at 10KHz a.k.a. 4.5KHz


Aliasing II


Dealing with aliasing

Anti-aliasing filters Before we sample a signal x(t) at fs we lowpass filter it through a filter of bandwidth B≤fs/2

This is straightforward but not simple 1.  An ideal lowpass filter has non-timelimited

non-causal response 2.  A timelimited signal has a Fourier transform of

unbounded support in frequency 3.  We can only approximately lowpass filter

1.  We need to worry about roll-off near fs/2

2.  We have to back off from fs/2

3.  Fast roll-off is accompanied by passband ripple

You cannot remove aliasing after sampling


Aliasing in practice

Any data-logging equipment worth a dime has anti-aliasing filtering built in

It is switched in at sample-rate selection time and is handled by the equipment software

DSpace, LabView, HP analyzers Often this (and timing questions) limits the available sampling rates High-performance data acquisition is an art The data quality near Nyquist is dodgy

So why do we need to know about aliasing? Because of the mistakes you can/might/will make!!


Downsampling scenario

An email arrives containing two columns of data sampled at 44.1KHz

You look at it and figure that it is oversampled and that most of the interesting information is in the range less than 10KHz in the signal

This would be captured by data sampled at 22.5KHz

So you toss out every second sample Now it is the same data sampled at half the rate

Let us try this out with a former Hilltop High student


Downsampling


Original data at 44.1KHz

Downsampled data at 22.05KHz

Decimated data at 22.05 KHz

Difference at 22.05 KHz Between downsample and decimate

Decimation instead of downsampling

The mistake earlier was that the reconstructed content of the 44.1KHz data contains signals in the range 0-20KHz roughly

Downsampling to 22KHz and then reconstructing should yield signals in the 0-10KHz range

The 10-20KHz material has been aliased into 0-10KHz

Before downsampling we should filter the data with a lowpass digital filter of bandwidth ~10KHz

This is called decimation


Matlab’s ‘decimate’


>> help decimate decimate Resample data at a lower rate after lowpass filtering. Y = decimate(X,R) resamples the sequence in vector X at 1/R times the original sample rate. The resulting resampled vector Y is R times shorter, i.e., LENGTH(Y) = CEIL(LENGTH(X)/R). By default, decimate filters the data with an 8th order Chebyshev Type I lowpass filter with cutoff frequency .8*(Fs/2)/R, before resampling. Y = decimate(X,R,N) uses an N'th order Chebyshev filter. For N greater than 13, decimate will produce a warning regarding the unreliability of the results. See NOTE below. Y = decimate(X,R,'FIR') uses a 30th order FIR filter generated by FIR1(30,1/R) to filter the data. Y = decimate(X,R,N,'FIR') uses an Nth FIR filter. Note: For better results when R is large (i.e., R > 13), it is recommended to break R up into its factors and calling decimate several times.

Matlab’s ‘downsample’

Downsampling is asking for trouble with physical signals It is a common mistake made by people

It is not really forgivable though No graduate of MAE143A has ever admitted to me to making this mistake


>> help downsample downsample Downsample input signal. downsample(X,N) downsamples input signal X by keeping every N-th sample starting with the first. If X is a matrix, the downsampling is done along the columns of X. downsample(X,N,PHASE) specifies an optional sample offset. PHASE must be an integer in the range [0, N-1].

Discrete Fourier transform Discrete-time Fourier transform (DTFT)

Function of continuous frequency Ω

Scaled to hide the sample time T dependence Periodic in Ω with period 2π Nice properties but not really useful

For timelimited discrete-time signal x[n], n=0,…,N-1 define the Discrete Fourier Transform (DFT)


X(⌦) =1X

n=�1x(nT )e�jn⌦ = Xs(!)|!=⌦

T

Xk =N�1X

n=0

x[n]e�jk 2⇡N n

, k = 0, 1, ..., N � 1

x[n] =1

N

N�1X

k=0

Xkejn 2⇡

N k, n = 0, 1, ..., N � 1

Discrete Fourier transform

DFT: N signal samples in and N transform values out Related to DTFT

Inverse DFT: N in values and N time samples out N is referred to as the record length If {x[n]} is real then


Xk =N�1X

n=0

x[n]e�jk 2⇡N n

, k = 0, 1, ..., N � 1

x[n] =1

N

N�1X

k=0

Xkejn 2⇡

N k, n = 0, 1, ..., N � 1

Xk = X(⌦)|⌦= 2⇡kN

X�k = X̄k

Interpretation of the DFT

N-sample data record

Periodic repetition – forever What is the Fourier series?

Period of this repetition is NT


{x0, x1, . . . , xN�1}x0

x1x2

xN�1

T 2T (N-1)T 0

time time

x0 x0x0

… … } } } }period NT period NT period NT period NT

xrep(t) =1X

k=�1cke

jk 2⇡NT t

ck =1

NT

Z NT�

0�xrep(t)e

�jk 2⇡NT t

dt

xrep(t) =

1X

n=�1x[n mod N ]�(t� nT )

Interpretation of the DFT

Repeat the N samples Periodic time-unlimited

Compute Fourier series – it is a periodic signal

This is the k-th DFT coefficient

ck is periodic in k with period N


xrep(t) =

1X

n=�1x[n mod N ]�(t� nT )

ck =1

NT

Z NT�

0�xrep(t)e

�jk 2⇡NT t

dt =1

NT

Z NT�

0�

N�1X

n=0

x[n]�(t� nT )e�jk 2⇡NT t

dt

=1

NT

N�1X

n=0

x[n]

Z NT�

0��(t� nT )e�jk 2⇡

NT tdt =

1

NT

N�1X

n=0

x[n]e�jk 2⇡N n

=1

T

Xk

The Discrete Fourier Transform

The DFT of an N-sample (real) signal record is an N element (complex) transform record is the Fourier series of the periodic (sampled) signal repeated over and over again in both directions is N-periodic in frequency bin (k)

We cannot resolve frequency more finely because there are only N data The frequency data have a conjugate relation Plotting |Xk| is a line spectrum as if the data were N-periodic


X�k = X̄k

Sunspot example from the Kamen & Heck book

Go to: http://sidc.oma.be/index.php3 Monthly sunspot data since 1749

Edit the file to remove leading and trailing rows with ‘*’ or fewer columns than four

>> load spot.dat

>> spd=spot(1:300,3);

>> plot(1749+[1:300]/12,spd)

300 data points 25 years

Compute the DFT


1750 1755 1760 1765 17700

20

40

60

80

100

120

140

160Sunspot data from Belgian Royal Observatory

year

num

ber

DFT of sunspot data

MAE143A Signals & Systems Winter 2015 53 0 50 100 150 200 250 300

0

2000

4000

6000

8000

10000

12000

14000

16000DFT magnitudes for 300 sunspot data

k

|Xk|

Note the symmetry about bin 150 Note strong dc component Note something at low frequencies

Zooming in on sunspot data

>> fdspot=abs(fft(detrend(spd))); stem([0:9],fdspot(1:10))

Bin 2

150 month cycle

Bin 3

100 month cycle

Real sunspot cycle

132 months


0 1 2 3 4 5 6 7 8 90

500

1000

1500

2000

2500

3000

3500Zoomed DFT magnitudes for 300 detrended sunspot data

k

|Xk|

Sunspot data DFT with 264 samples


0 1 2 3 4 5 6 7 8 90

500

1000

1500

2000

2500

3000

3500

4000

4500

5000Zoomed DFT magnitudes for 264 detrended sunspot data

k

|Xk|

DFT

The DFT allows analysis of a data record by computing the Fourier series as if the data record were one period of an infinite periodic sequence

This is good for capturing periodic phenomena in the data provided you have an integer number of periods in the data

It is easy to be misled Especially if you forget the above explanation

The DFT is a fundamental tool in data analysis There is a huge number of helper functions in matlab


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Fourier Analysis & Discrete-time Systems - …numbat.ucsd.edu/~bob/signals2015Winter/Signals_&_Systems_2015... · Discrete-time signals & systems ... Direct solution ... Moving from