signal processing – summarygrezl/zre/lectures/02_sig_en.pdf · analog to digital (ad) converter...

Signal Processing – Summary

Jan Cernocky, Valentina Hubeika

cernocky,[email protected]

DCGM FIT BUT Brno, [email protected]

FIT BUT Brno

Signal Processing – Summary Jan Cernocky, Valentina Hubeika, CDGM FIT BUT Brno 1/41

Agenda

• Introduction

• Analog signals and sampling

• Frequency analysis of sampled signals.

• Random signals.

• Filters.


Why digital signal processing ?

• reproducibility (we don’t have to think about “component tolerance”).

• no changes due to material aging or temperature.

• easy setup (may be difficult tuning for analog signals).

• possible adaptive processing (“changing device functionality depending on the type of

the signal”).

• simulation = application.


Usual Approach

A/Ddigitalprocessing(PC, DSP)

D/A

storingtransmisioninterpretationother processing...

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..................................................

x(t) x (n) y (n) y(t)

The input signal is continuous: function of a real-valued variable (in our case time) t, t is

defined over the interval (−∞, ∞), t → ∞.


t

s(t)

0

To represent the signal in frequency domain (spectrum), Fourier transform is used:

X(f) =

∫ −∞

−∞

x(t)e−j2πftdt, (1)

where the function X(f) is called the spectral function, or in short spectrum. The

function is defined ∀f from −∞ to ∞ and is complex. Its magnitude is |X(f)| and its

phase is 6 X(f). Here, we talk about magnitude and phase spectra.


In case of real signals, we will need only the right side of the spectral function (f > 0).

The left side is complex adjoint to the right side:

X(f) = X⋆(−f), (2)

or |X(f)| = |X(−f)| a arg X(f) = − arg X(−f).

f max0-f max f

|S(f)|

⇒ Good signals are frequency constrained (energy within the band of (0, fmax) ).

⇒ Spectral function cant be calculated (infinite, integral,. . . ).


Analog to Digital (AD) Converter

sampledsignal xs(n)

- -

££

££° -

£££°- antialias.

filtersampling quantization

quantizedsignal xq(n)

x(t)

Sampled signal is calculated by multiplying the original signal with a periodic function

over time.

sD

x (t)=x(t)s(t)

t

s(t)

0t0 1/D T

Ds(t)


Theoretically, sampling is multiplying the original analog signal with a series od Dirac

deltas (unit impulse function) (infinite height, zero width, unit area). The result of

multiplication is again a series of Dirac impulses, with the powers corresponding to the

values of the original signal at times nT :

1 1

s

t0 T

s(t)D

x (t)=x(t)s(t)

t

s(t)

0

T is the sampling period

Fs =1

Tis the sampling frequency


The spectrum of the sampled signal is periodized !!!

Xs(f) =1

T

+∞∑

n=−∞

X(

f −n

T

)

=1

T

+∞∑

n=−∞

X (f − nFs) (3)

From the point of view of the sampling theorem:

1) Fs > 2fmax: The copies of the spectrum of the original signal do not overlap so the

original signal can be perfectly reconstructed from its sampled version by filtering it with

low-pass filter with the frequency Fs/2.

2) Fs ≤ 2fmax: the copies of the spectrum of the original signal do overlap so the

resulting spectrum has a different shape than the original spectrum. The original signal

cannot be reconstructed from its sampled version. We observe so called aliasing.

Shanon–Kotelnikov–Nyquist–sampling theorem

Fs > 2fmax


1. Example: sampling and reconstruction – OK

Fs = 8000 Hz, fmax = 3000 Hz, so that Ωs = 16000π rad/s, ωmax = 6000π rad/s.

T = 18000 s

−1.5 −1 −0.5 0 0.5 1 1.5

x 105

0

0.5

1

−1.5 −1 −0.5 0 0.5 1 1.5

x 105

0

2000

4000

6000

8000

−1.5 −1 −0.5 0 0.5 1 1.5

x 105

0

0.5

1

x 10−4

−1.5 −1 −0.5 0 0.5 1 1.5

x 105

0

0.5

1


2. Example: sampling and reconstruction – BAD

Fs = 8000 Hz, fmax = 7000 Hz, so that Ωs = 16000π rad/s, ωmax = 14000π rad/s.

T = 18000 s

−1.5 −1 −0.5 0 0.5 1 1.5

x 105

0

0.5

1

−1.5 −1 −0.5 0 0.5 1 1.5

x 105

0

2000

4000

6000

8000

−1.5 −1 −0.5 0 0.5 1 1.5

x 105

0

0.5

1

x 10−4

−1.5 −1 −0.5 0 0.5 1 1.5

x 105

0

0.5

1


Antialiasing filter – constrain to [−Fs/2, Fs/2]

−1.5 −1 −0.5 0 0.5 1 1.5

x 105

0

0.5

1

−1.5 −1 −0.5 0 0.5 1 1.5

x 105

0

0.5

1

−1.5 −1 −0.5 0 0.5 1 1.5

x 105

0

5000

−1.5 −1 −0.5 0 0.5 1 1.5

x 105

0

0.5

1

x 10−4

−1.5 −1 −0.5 0 0.5 1 1.5

x 105

0

0.5

1


Denotation of the Sampled Signal

xs(nT ) or just xs[n] — the sampled signal is a series of values.

1. Along with the sampled signal, we always need information on the sampling frequency

(implicit or explicit)

2. When working with a sampled signal, we do not want to consider the sampling

frequency in the calculations, therefore assume that the period is T ′ = 1, thus F ′s = 1.

The normalized time is then given as:

t′ =t

T, thus n =

nT

T(4)

The normalized frequency is:

f ′ =f

Fs

(5)


Example

Define a function generating cosine with the frequency of 200 Hz with the sampling

frequency Fs = 8000 Hz.

In the continuous time: s(t) = cos(2πf0t) = cos(2π200t).

While sampling we will substitute the continuous time variable t with the discrete time nT ,

where T is the sampling period: x(nT ) = cos(2πf0nT ) = cos

(

2πf0

Fs

)

n.

The frequencyf0

Fs

is normalized. The desired signal is defined as: x(n) = cos

(

2π1

40n

)

.

Generate 1 second of such a signal in Matlab:

n = 0:7999;

x = cos (2 * pi * 1 / 40 * n);

wavwrite(x,8000,16,’sig.wav’);


Behaviour of the Sampled Signal in Frequency Domain — Spectrum

Discrete Fourier transform – DTFT – definition:

X(k) =

N−1∑

n=0

x[n]e−j2π nkN pro k ∈< 0, N − 1 > (6)

How to use it for discrete signals?:

• analyze a “window” of the length of N samples.

• result: when multiplying values X(k) with the sampling perioud T , the approximation

of the spectral function at the points k∆f , where ∆f =Fs

N(true frequency) or

∆f ′ =1

N(normalized frequency (usually not used)

X(k∆f) = T

N−1∑

n=0

x[n]e−j2π nkN (7)

in this definition, x[n] can be substitute for x(nT ).


Comparing to the spectral function of the corresponding analog signal, the spectrum of the

discrete signal is not the same thing!

1. we calculate spectrum of the sampled signal, so the spectrum is periodic with the

period N units (corresponds to the sampling frequency Fs). If k ∈ (−∞, +∞), then

X(k∆f) is repeated after each N values.

2. The signal is windowed. The resulting spectrum carries information on the window:

the signal is multiplied with the window function in time, the spectrum of the window

is then convoluted with the spectrum of the original signal. This often results in

blurring of the theoretically sharp lines of the spectrum (for instance during analysis of

the harmonic signal). More in the lecture on preprocessing.

3. if the spectrum is discrete (we dispose of only N values from 0 to Fs), then the signal

must be periodic !

time frequency

sampling periodization

periodization discretization


How can we use it in practice?

We want to make frequency analysis of one voiced frame of speech

s = wavread(’test.wav’)’;

sfr = frame (s,160,80);

x = sfr(:,13);

plot (x);

0 20 40 60 80 100 120 140 160−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4


DTFT only – careful with the frequency axes:

Fs = 8000; f = (0:159) / 160 * Fs; X = fft(x);

subplot (211); plot(f,abs(X)); subplot (212); plot(f,angle(X));

0 1000 2000 3000 4000 5000 6000 7000 80000

1

2

3

4

5

6

7

0 1000 2000 3000 4000 5000 6000 7000 8000−4

−2

0

2

4


DTFT only – the left half is symmetric to the right half (we consider only on in analysis)

Fs = 8000; f = (0:79) / 160 * Fs; X = fft(x); X = X(1:80);


0 500 1000 1500 2000 2500 3000 3500 40000

1

2

3

4

5

6

7

0 500 1000 1500 2000 2500 3000 3500 4000−3

−2

−1

0

1

2

3

4


If we want to smoother the spectrum (add more points – shorten intervals between values),

having only one frame (without a possibility of extension), we can use ⇒ zero padding

Fs = 8000; f = (0:511) / 1024 * Fs;

X = fft([x’ zeros(1,1024-160)]); X = X(1:512);


0 500 1000 1500 2000 2500 3000 3500 40000

2

4

6

8

0 500 1000 1500 2000 2500 3000 3500 4000−4

−2

0

2

4


FREQUENCY ANALYSIS OF RANDOM SIGNALS

Speech signal is a random signal. Thus we use power spectral density (PSD) function

for its frequency analysis. PSD is real and represent power density of the signal in

frequency domain. One of the possible ways of computation uses DTFT:

GDTFT (k∆f) =1

N|X[k]|2.

Which is just the power of the magnitude.

⇒ careful with |X[k]|2. in Matlab – power of the complex number is not the same as the

power of its magnitude:

• X = fft(x); Gdft = X .^ 2; wrong !

• X = fft(x); Gdft = abs(X) .^ 2; good !

• X = fft(x); Gdft = X .* conj(X); good and fast ,


Example:

Fs = 8000; f = (0:511) / 1024 * Fs

X = fft([x’ zeros(1,1024-160)]); X = X(1:512); Gdft= 1/160 *abs(X) .^ 2;

plot(f,Gdft);

0 500 1000 1500 2000 2500 3000 3500 40000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Dynamic performance of the power spectral density is higher than of DTFT (second

power) which causes poor visualization of “weak” parts of the image. For this reason,

often decibels are used (Matlab: function log10)

GDTFT (k∆f) = 10 log10

1

N|X[k]|2.

0 500 1000 1500 2000 2500 3000 3500 4000−80

−70

−60

−50

−40

−30

−20

−10

0


LINEAR FILTERS

filter

x(n) y(n)

Usually, filters are

• linear — hold linear combination : if x1(n) → y1(n) a x2(n) → y2(n), then

ax1(n) + bx2(n) → ay1(n) + by2(n), where a, b ∈ ℜ.

• time invariant — behaviour does not depend on time: if x(n) → y(n), then as well

x(n − n0) → y(n − n0), where n0 is any arbitrary shift. Sometimes however we desire

filters with time variant behaviour — adaptive systems, speech frames (change

∀10 ms).

• causal — the filter “does not see into future”: y(n) ∼ y(m < n) a x(m ≤ n).


Impulse Response

or impulse characteristics is the response of a filter to the Kroneck or unit impulse (do not

confuse with the Dirac delta):

δ(n) =

0 pro n 6= 0

1 pro n = 0

(8)

filter

(n)δ h(n)

Knowing the impulse response, we can calculate the response to any input signal. Each

sample is perceived as a shifted and scaled unit impulse. We represent the output as the

convolution:

y(n) = x(n) ⋆ h(n) =∞∑

m=−∞

x(m)h(n − m) =∞∑

m=−∞

h(m)x(n − m) (9)


Analysis of the impulse characteristics:

• if h(k) = 0 for ∀k < 0, then the filter is causal (all samples after n-th sample will be

multiplied by zero).

• impulse response can be finite — FIR (finite impulse response) or infinite — IIR

(infinite impulse response).

• the Fourier image in frequency then represents the complex characteristics of the filter:

h(k) → H(f)

Convolution in time domain correspond to product in frequency domain, thus the

spectrum of the resulting signal is:

Y (f) = X(f)H(f) (10)

We should keep in mind that we work with the discrete signals (so is the impulse response

of a filter) the image in the frequency is therefore periodic with the period Fs or 1 for the

normalized frequency.


Representation of a Filter

z -1 z -1 z -1...

bQ-1

z -1 z -1 z -1Σ

...

-a

-a

1

2

bQ

...

bQ-2

-a Pb0

-ab1 P-1

...

y(n)

x(n)

The block z−1 denotes a delay of 1 sample. The behaviour of a filter can be expressed as

a difference equation:

y(n) =

Q∑

k=0

bkx(n − k) −

P∑

k=1

aky(n − k), (11)


where x(n− k) are actual and delayed input values and y(n− k) are delayed output values.

Types of filters:

• FIR – non-recursive: only b0 . . . bQ nonzero. Always stable.

• IIR – solely recursive: only b0, a1 . . . aP nonzero.

• IIR – generally recursive: ai and bi nonzero.

It is difficult to estimate the behaviour of the filter and analyze it from the point of view of

stability from the difference equation.


Z-Transform

X(z) =

∞∑

n=−∞

x(n)z−n (12)

The z-transform is a generalization of the DTFT. The z-transform becomes the DTFT

when z = expjw (when DTFT is evaluated on the unit circle). In order to determine the

frequency response of the system the z-transform must be evaluated on the circle (the

region of convergence must contain a circle). Otherwise, the DTFT of the system does not

exist.

Lets define delay as: if x(n) → X(z), then for y(n) = x(n − n0) will hold:

Y (z) = z−n0X(z) (13)

for the delay of one time unit holds: x(n − 1) → z−1X(z). Thus the delay of one time

unit will be denoted as:

z -1


Transfer Function of a Filter

We can rewrite the difference equation using z-transform

Y (z) =

Q∑

k=0

bkX(z)z−k −P

∑

k=1

akY (z)z−k, (14)

The transfer function can be defined as the fraction:

H(z) =Y (z)

X(z)=

Q∑

k=0

bkz−k

1 +

P∑

k=1

akz−k

=B(z)

A(z), (15)

where B(z) and A(z) are two polynomes. The a0 coefficient of the denominator must be

equal to 1. a0 does not physically present in the filter (denotes that the filter has the

output).


Frequency Characteristics

of a filter from 0 to Fs (or from 0 to 1 in using normalized frequency) can be acquired

from the transfer function by analyzing the unit circle. We are interested in the complex

values of the function H(z):

H(f) = H(z)|z=ej2πf (16)

for the normalized frequency. Or:

H(f) = H(z)|z=ej2πfT (17)

for the normal frequency.

For each value f we need to calculate the position of the point on the unit circle:

z = ej2πf (complex number), then for this number we need to compute the fraction of the

polynomes B(z) and A(z) (complex number). In Matlab by using function freqz.


Re

Im

1~Fs1/2~Fs/2

"z"

πexp(j2 f)

0 1

−j

−1

+j


Zeros and Poles of the Transmission Function. . .

The other way how to represent the transfer function H(z) is using product:

H(z) =B(z)

A(z)=

b0 + b1z−1 + . . . + bQz−Q

1 + a1z−1 + . . . + aP z−P

=z−Q(b0z

Q + b1zQ−1 + . . . + bQ)

z−P (zP + a1zP−1 + . . . + aP )

=

= b0z−Q

z−P

Q∏

k=1

(z − nk)

P∏

k=1

(z − pk)

= b0zP−Q

Q∏

k=1

(z − nk)

P∏

k=1

(z − pk),

If ak, bk ∈ ℜ, then poles pk and zeros nk can be either real or complex conjugate in pairs.

Filter is stable when all poles lie within the unity circle:

|pk| < 1


Example of a Filter

We want to implement a filter that will be simulating a telephone channel to filter a signal

of the CD quality. Our filter will be a band-pass filter from 300 to 3400 Hz. There are

several functions available in Matlab to implement a filter. We will be using so called

elliptic filters:

Fs = 44100; Fs2 = Fs/2; % has to be normalized by half Fs

Wp = [300/Fs2 3400/Fs2]; % pass-band

Ws = [200/Fs2 3500/Fs2]; % stop-band - approximately

Rp = 3; % ripple in pass-bandu dB

Rs = 30; % muting stop-bandu dB (obe hodnoty od

% see the norms.

[N, Wn] = ellipord(Wp, Ws, Rp, Rs) % computing of the filter order

[B,A] = ellip(N,Rp,Rs,Wn) % computing of polynoms B a A

. . . as the result we have 2 polynomes of the 12th order.


Frequency characteristics: freqz (B,A,512,Fs);

0 0.5 1 1.5 2

x 104

−1500

−1000

−500

0

Frequency (Hz)

Pha

se (

degr

ees)

0 0.5 1 1.5 2

x 104

−100

−50

0

50

Frequency (Hz)

Mag

nitu

de (

dB)


Poles and zeros: zplane (B,A);

−0.5 0 0.5 1 1.5 2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Real Part

Imag

inar

y P

art


Implementation in C

• the basic implementation of the direct structure is very simple – just rewrite the

difference equation into C (see the file filter.c)

• in practice, usually optimal structures are used, which have only one delay bus and are

not prone to the rounding errors.

• for more information on the filter theory see SXC/ISS – lecture “discrete systems”:

http://www.fit.vutbr.cz/~cernocky/sig


Passing a Random Signal through a Filter

Let the filter have a complex frequency characteristic H(f). For the input signal with the

spectral power density Gx(f) the output spectral power density is given as:

Gy(f) = |H(f)|2Gx(f)

. . . the input PSD is multiplied by the power of the magnitude of the complex frequency

characteristic.


Example: Given a realization of a random signal. The

signal is filtered by the filter defined as H(z) = 1−0.9z−1. The output ignal and its PSD are:

50 100 150 200 250 300

−0.2

0

0.2

0 0.5 1 1.5 2 2.5 3

0.1

0.2

0.3

0.4

0.5


The magnitude of the frequency characteristic and its power are:

0 0.5 1 1.5 2 2.5 3

0.5

1

1.5

|H|

0 0.5 1 1.5 2 2.5 3

0.5

1

1.5

2

2.5

3

3.5

|H|2


The output signal and its PSD:

50 100 150 200 250 300

−0.2

0

0.2

0 0.5 1 1.5 2 2.5 3

0.02

0.04

0.06

0.08

0.1


signal processing – summarygrezl/zre/lectures/02_sig_en.pdf · analog to digital (ad) converter...

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