Signal analysis & data processing – Spectral leakage & window functions

Exercise 7
Spectral analysis using DFT (FFT). Spectral leakage
and window functions.

Complex Fourier transformation pair of periodic signals

k 
 2kt 
INVERSE: x( t )   Xk exp i
 T


 F-1[X()]
k  

T/2
1  2kt 
DIRECT: Xk 
T  x( t ) exp  i

 dt
T 
 F[x(t)]
T / 2

X k  ak  i  b k

(direct) Fourier transformation

F[x(t)]

F-1[X()]
inverse Fourier transformation

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 Signal analysis & data processing – Spectral leakage & window functions

1. Let's sample integer (complete) number of sine wave periods

T = kTs = 3 Ts (k – integer)

2 4  6 
spectral domain:   0, , , (   s ),…
T T T
2. Let's sample the incomplete number of sine wave periods
T  kTs (k – integer)

x(t)
Ts 2Ts < T < 3Ts
T/2 > Ts > T/3

4 6
 s 
T T
t 2 < s < 3

T=t .N

What happens ???

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 Signal analysis & data processing – Spectral leakage & window functions

Parseval theorem for DFT

temporal domain:


Nt  E x 2 (t)   
 E x r2 

spectral domain:
i  N 1


2
N  X(i )
i0

equality of the values of power determined in both domains

Nt = N
conclusion:
2 4 6
power must be transferred to from    s to   0, , , ,…
T T T

LEAKAGE PHENOMENON

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 Signal analysis & data processing – Spectral leakage & window functions

What is the origin of leakage ?

DFT multiplies signal seen within a window of observation leading

to signal discontinuities - its actual distortion

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 Signal analysis & data processing – Spectral leakage & window functions

solution - application of smoothing functions (windows)

windows applied:

useful formulas: 1B = log (P/Pref)

1dB = 1/10 B
PdB = 10 log (P/Pref); P = 2 = A2/2
PdB = 10 log ((/ref)2) = 20 log (/ref)

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 Signal analysis & data processing – Spectral leakage & window functions

Compulsory tasks
a) analyse how the spectrum bandwidth (frequency range) and
resolution may be controlled,
b) determine spectra for the following signals:
- monoharmonic,
- polyharmonic (e.g. square wave),
- noise,
- compound (e.g. sine wave + noise),

Be careful about presenting the entire spectrum (select

 relevant maximum frequency)

c) analyse the influence of an incomplete number of sine wave

periods on its spectrum  spectral leakage

d) apply various smoothing functions (windows) for the reduction

of leakage - perform comparative study and choose the best
window taking into account:
 height and bandwidth of the main lobe,
 side-lobe fall-off.

Consider the presentation of spectra in logarithmic scale (in

 dB) to better show the leakage

Optional tasks

- analyse the distribution of the leaked spectrum of the single sine

wave by changing the number of periods (frequency) between N
and N+1 in several steps

- try to reduce (avoid) leakage by the change of sampling

frequency

- .................

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 Signal analysis & data processing – Spectral leakage & window functions

The exercise aims at learning how to:


 how the parameters of spectral analysis should be
selected (range, resolution, scale type),
 how to recognise the signal type and its parameters
from the spectrum,
 how to avoid/reduce leakage using windows.

project "DFT - windows" in LabView environment

1B = log (P/Pref)
1dB = 1/10 B
PdB = 10 log (P/Pref); P = 2 = A2/2
PdB = 10 log ((/ref)2) = 20 log (/ref)

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 Signal analysis & data processing – Spectral leakage & window functions

Sample problems
1. What is the minimum number of samples for the period of harmonic
allowing for correct reconstruction of the frequency ?
2. Formulate the Shannon (Nyquist) condition for sampling.
3. What is the origin of aliasing ?
4. What is the origin of spectral leakage ?
5. How can the aliasing be avoided ?
6. How does the smoothing function (window) work ?
7. Describe the way the signal is reconstructed from its amplitude
(amplitude-phase) spectrum.
8. How is the spectral resolution related to signal properties ?
9. What is the spectrum range for DFT ?
10. Define a decibel (dB).
11. What is the reduction of the signal's amplitude corresponding to
-3dB, -20dB, -40dB ?
12. What are the main and side lobes of the window ?
13. What is the width of the main lobe for the rectangular window ?
14. What's the side lobe roll-off and how is it defined ?
15. What is the difference between amplitude and power spectra ?
16. What are the range and the resolution of the spectrum if the following
data are given:
 signal type - sine wave
 signal frequency - 50Hz
 signal amplitude - 1V
 sampling frequency - 2000Hz
 No of samples - 1000
17. What are the range and the resolution of the spectrum if the following
data are given:
 signal type - sine wave + random noise
 sine wave frequency - 25 Hz
 sine wave amplitude - 1V
 noise variance - 2 V2
 signal duration - 1s
 No of samples - 2000