dsp first, 2/e reading assignmentsdspfirst.gatech.edu/archives/lectures/dspfirst-l19-pp4.pdf ·...

DSP First, 2/e

Lecture 19S SSpectrogram: Spectral Analysis via DFT & DTFT

READING ASSIGNMENTSREADING ASSIGNMENTS

This Lecture: Chapter 8, Sects. 8-6 & 8-7

Other Reading:Other Reading: FFT: Chapter 8, Sect. 8-8

Aug 2016 © 2003-2016, JH McClellan & RW Schafer 3

LECTURE OBJECTIVESLECTURE OBJECTIVES

Spectrogram Spectrogram

Time Dependent Fourier Transform Time-Dependent Fourier Transform DFTs of short windowed sections

Frequency Resolution To resolve closely spaced spectrum lines Need long windows


Review & RecallReview & Recall Discrete Fourier Transform (DFT)Discrete Fourier Transform (DFT)

1

)/2(][][N

nkNjenxkX

1

)/2(][1][N

nkNjekXN

nx

DFT is frequency sampled DTFT

0

][][n

0

][][kN

q y p For finite-length signals

DFT computation is actually done via FFT FFT of zero-padded signalmore freq samples

Transform pairs & properties (DTFT & DFT)Aug 2016 © 2003-2016, JH McClellan & RW Schafer 5

Transform pairs & properties (DTFT & DFT)

Recall: DFT and DTFTRecall: DFT and DTFT

1

)/2(][][N

knNjenxkX Discrete Fourier

11 N

0

)(][][n

jenxkXTransform (DFT)

1

0

)/2(][1][N

k

knNjekXN

nx Inverse DFT

njj enxeX ˆˆ ][)(Discrete-time Fourier Transform (DTFT)

n

2 ˆˆ ˆ)(1][ deeXnx njjInverse DTFT


0

)(2

][ deeXnx

The Transform Method (DTFT)( )

To get the output signal y[n] given the input x[n] andTo get the output signal y[n], given the input x[n], and the system defined by h[n].

][][][ nxnhny

)(][ ˆeXnx j

][][][y

][)()()()(

)(][

ˆ

ˆˆˆ

YeXeHeY

eXnx

j

jjj

Convolution becomes multiplication of DTFTs

][)( nyeY j


Convolution becomes multiplication of DTFTs

Frequency Response MethodFrequency Response Method

To get the output signal y[n] given the input x[n] is aTo get the output signal y[n], given the input x[n] is a sinusoid, and the system defined by h[n].

)ˆcos(][][ nAnhny

)(][

)cos(][][

ˆ

0

eHnh

nAnhny

j

)()( eval 0ˆˆ 0 HeHeH jj

M lti l M it d Add Ph

)ˆcos(][ 000 HnHAny


Multiply Magnitudes, Add Phases

DTFT, DFT and DFS are Tools for AnalysisTools for Analysis• DTFT applies to discrete time sequences regardless of length (as• DTFT applies to discrete time-sequences, regardless of length (as

long as x[n] is absolutely summable);• Note: sinusoids are not absolutely summable

• N-pt DFT of a finite-length sequence is equivalent to sampling the corresponding DTFT on frequency axis at N equally spaced

i t ith t l i i f tipoints, without losing any information• Zero-padding when N>L

• If the sequence is periodic, and the DFT is performed on one period, the result is a discrete Fourier Series (DFS) which is an exact representation.


Various Situations in Signal AnalysisSignal AnalysisA Signal property does not fluctuate with timeA. Signal property does not fluctuate with time

1. Sampled signal is periodic with known period N0 – use N0-pt DFT and get the precise result (DFS)

2 Sampled signal is aperiodic or period is unknown or hard toCOMMON 2. Sampled signal is aperiodic, or period is unknown, or hard to ascertain precisely.

3. Signal characteristic is global (for the entire signal)a. Finite-Length, e.g., pulse

CASE

g , g , pb. Infinite-Length: e.g., right-sided exponential

B Sampled signal property changes with timeB. Sampled signal property changes with time1. Signal defined by math formulas over intervals2. Recorded signal, most likely with changing sinusoidal content, or

ti th t lCOMMONCASE


properties – the most general caseCASE

ExamplesExamples

A 1 P i di20,100),50cos()5.020cos(2 0 Nftt s

A.1 Periodic Sequence

100),150cos()5.020cos(2 fttA.2 Nonperiodic

Sequence

100),150cos()5.020cos(2 sftt

A.3 Global infinite-length

,....2,1),cos(3 1.0 nne n

gsequence

B.2 A general


sequence

A.3: DTFT gives globalspectrumspectrum

21)cos(3 1.0 nne n A.3 sequence from formula

,....2,1),cos(3 nne

N-pt DFT: approaching DTFTwhen N infinity

0 2


More about this later.

When Periodicity Is Not Precise A 2Precise – A.2 Periodicity is imprecise – either not known exactly or y p y

impractical to take exactly one period of data; for example:

16L

200 N Actual period of sequence

),50cos()5.020cos(2)( tttx

We take L data points for analysis; actual period is not L

20,100,5 00 Nff s


p y ; p(20 for the example).

Imprecise Period Example (2)

16L

)50cos()5020cos(2][ nnnx 200 N

16L

1ˆ2.0102

642)5.6(

),5.0cos()5.02.0cos(2][ nnnx

2ˆ5.0642)16(

64N|][| kX

64N16N

0 6 16 48 58 63

N32N16L

|)(| jeX

0 6 16 48 58 63


Similar Example – Changing Lp g g)5.1cos()5.02.0cos(2 nn

|][| kX16L1ˆ2.0

102

642)5.6(

ˆ512)16( |][| kX

0 6 16 48 58 63 64N

25.1264

)16(

|)(| jeX

ˆ

NL 16

|)(| jeX1ˆ2.0

2ˆ5.1 8.1ˆ L 320

512ˆ


|)(| eX

2

N 5.12

ObservationsObservations When signal property stays fixed the longer the section of data When signal property stays fixed, the longer the section of data

(larger L), the sharper the frequency resolution.

The larger the number of points for the DFT N the denser the The larger the number of points for the DFT, N, the denser the frequency axis sampling.

However the resolution (or separation) of sinusoids w r t frequency However, the resolution (or separation) of sinusoids w.r.t. frequency is determined by the data length L, not by length of the N-pt DFT (with zero padding).

When signal properties change with time, there are other considerations that limit the size of L; see case B.


Case B The General CaseCase B – The General Case

DFT can also be used as a tool for finding DFT can also be used as a tool for finding the spectral composition of an arbitrary signal:signal: In general, we observe changing signal

propertiesproperties Short-time analysis framework Analyze sequence one block (frame) of data at a Analyze sequence one block (frame) of data at a

time using DFT Successive blocks of data may overlap


“SPECTROGRAM”

FRAME = WINDOW of DATAFRAME WINDOW of DATA

FT

frame thiframe )1( thi

Example of Non-overlapped frames

frame)2( thi

Data also can be extracted from overlappedframes – overlapped characteristics provide smoother transitions from one frame to nextN


XNDenote Number of Overlapping Points as XN

A very long signalA very long signal

F i t l Four intervals Four spectrum lines

2999500)1110cos(24990)211.0cos(5

nnnn

99995000)40(49993000)8.0cos(2

2999500)111.0cos(2)/(][

1

nnnn

fnxnx s

99995000)4.0cos(21 nn


DFT of a very Long Signal (Global)(Global) Length 10,000 signal, 16384-pt DFTLength 10,000 signal, 16384 pt DFT


Windowing a Long SignalWindowing a Long Signal Long signal is time-shifted into the domain of the window, g g ,

[0,300] in this case


Von Hann Windowin Time Domainin Time Domain

Plot of Length-20 von Hann windowg

L00

)(LengthWindow Hannvon

LnLn

nnwh 0))1/()1(2cos(

00][ 2

121


Ln0

DTFT of Hann Window: Change LengthChange Length DTFT (magnitude) of windowed sinusoid.( g )

Length-20 Hann window vs. Length-40 Hann window

crossing-Zero

L 8ˆ

WidthPeak g


Short DFTs of Windowed Sections (L=301)Sections (L=301) 301-pt Hann window. 1024-pt DFT (zero-padded)


Short DFTs of Windowed Sections (L=75)Sections (L=75) 75-pt Hann window. 1024-pt DFT (zero-padded)


A very long signalA very long signal

F i t l Four intervals Four spectrum lines

2999500)1110cos(24990)211.0cos(5

nnnn

99995000)40(49993000)8.0cos(2

2999500)111.0cos(2)/(][

1

nnnn

fnxnx s

99995000)4.0cos(21 nn


Two Spectrograms ofvery Long Signalvery Long Signal

Window Lengths of L=301 and L=75; overlap 90%


MATLAB SpectrogramMATLAB Spectrogram• In MATLAB, useIn MATLAB, use

• Fs is the sampling frequency specified in samples/s• WINDOW:

S = spectrogram(X,WINDOW,NOVERLAP,NFFT,Fs,‘yaxis’)

sf• WINDOW:

- if a vector, extract segments of X of equal length, and then multiply window vector and data vector, element by element;

L

- If an integer, same as above but use a Hamming window of the specified length

- if unspecified, use default• NOVERLAP: (<WINDOW)

- Number of overlapping data between successive frames• NFFT: Length of FFTN

XN


g

Resolution TestResolution Test Closely spaced frequency componentsy p q y p How close can the lines be? Depends on window (section) length Inverse relationship

49990)227.0cos(5.1)2.0cos( nnn

99997000)7.0cos()6.0cos(369995000)2.0cos(

49990)227.0cos(5.1)2.0cos()/(][

nnnnn

nnnfnxnx s

)()(

80/810ˆ)70()60(3/?8027.0ˆ)227.0cos(5.1)2.0cos(

nn 296

18

ˆ )(


80/81.0)7.0cos()6.0cos(3 nn

Two Spectrograms:Resolution TestResolution Test

Window Lengths of L=301 and L=75, overlap 90%


Short DFTs ofWindowed SectionsWindowed Sections

75-pt and 301-pt Hann windows. 1024-pt DFT (zero-padded)


dsp first, 2/e reading assignmentsdspfirst.gatech.edu/archives/lectures/dspfirst-l19-pp4.pdf ·...

Documents