Lec 2 - Discrete Time Fourier Transform

Representation of Sequences by Fourier Transforms Many sequences can be represented by a Fourier integral of the form as

Discrete Time Fourier Transform

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Representation of Sequences by Fourier Transforms

H (e

)=

∑ h[n]e

n = −∞

− j ( ω + 2π ) n

=

∞

∑ h[n]e

− jωn

π

∫ π X (e

jω

−

X (e jω ) =

)e jωn dω.

∞

∑ x[n]e

− jωn

n = −∞

h[n ] =

1 2π

π

∫ π H (e

jω

−

)e jωn dω.

∞

H (e jω ) =

∑ h[n]e

− jωn

n = −∞

Fourier Transform (Convergence)

The frequency response of discrete-time LTI system is always a periodic function with period 2π. ∞

1 2π

Frequency response of a LTI system is simply the Fourier transform of the impulse response.

Maryam Mahsal Khan (Lecturer)

j ( ω + 2π )

x[n ] =

jω

= H (e )

n = −∞

More generally, H ( e j (ω + 2πr ) ) = H ( e jω ), for r an integer.

Determining the class of signals that can be represented Fourier transform is equivalent to considering the convergence of the infinite sum of the Fourier transform. A sufficient condition for convergence can be found as

X ( e jω ) =

∞

∑ x[n]e

n = −∞

− jωn

≤

∞

∑ x[n ] e

n = −∞

− jωn

≤

∞

∑ x[n] < ∞

n = −∞

Thus, if a sequence is absolutely summable, then its Fourier transform exists. The series can be shown to converge uniformly to a continuous function of ω. Since a stable sequence is, by definition, absolutely summable, all stable sequences have Fourier transforms.

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Fourier Transform (Interpretation)

Example 2.19 Ideal Frequency-Selective Filters: Ideal Lowpass filter

Signals: The Fourier Transform X ( e jω ) of a signal x[n] describes the frequency content of the signal. „

„

jω At each frequency ω0 , the magnitude spectrum X ( e 0 ) describes the amount of that frequency contained in the signal.

At each frequency ω0 , the phase spectrum ∠X (e jω0 ) describes the location (relative shift) of that frequency component of the signal.

Systems: The frequency response H ( e jω ) of a linear system describes how frequencies input to the system are modified: „

„

An input frequency component ω 0 is amplified or attenuated jω by a factor H (e 0 ) . An input frequency component ω 0 is shifted by an amount ∠H (e jω0 ).

DTFT of Ideal Low-Pass Filter

Contd.. Convergence of the Fourier Transform The oscillatory behaviour – Gibbs Phenomena

( ) ∑ sinπωn ne

H M e jω =

M

c

− jω

n=− M

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Frequency-Domain Representation of Discrete-Time Systems Eigenfunctions for LTI system „

Frequency-Domain Representation of Discrete-Time Systems (Cont’d) Frequency response of the system is defined as

Consider an input sequence

H ( e jω ) =

x[ n ] = e jωn for − ∞ < n < ∞,

The corresponding output of a LTI discrete-time system with impulse response h[n ] is

y[n ] = x[n ] * h[n ] =

∞

∑ h[k ]e ω

j (n−k )

k = −∞

H ( e jω ) =

„

If we define

„

Then we have

∞

∑ h[k ]e

⎛ ∞ ⎞ = e jωn ⎜ ∑ h[k ]e − jωk ⎟. ⎝ k = −∞ ⎠

− jωk

∞

∑ h[k ]e

− jωk

,

k = −∞ „

Real and imaginary representation

H ( e jω ) = H R ( e jω ) + jH I ( e jω )

,

k = −∞

y[n ] = H ( e jω )e jωn .

„

Magnitude and phase polar representation jω

„

H ( e jω ) = H ( e jω ) e j∠H ( e ) .

jωn

We define e as an eigenfunction of the system, and the associated eigenvalue is H ( e jω ) .

Example 2.17 Frequency Response of the Ideal Delay Consider the ideal delay system defined by

Example 2.20 Frequency Response of the Moving-Average System The impulse response of a moving-average system is

y[n ] = x[n − nd ], where nd is a fixed integer. „ „

1 ⎧ − M1 ≤ n ≤ M 2 , ⎪ h[n ] = ⎨ M 1 + M 2 + 1 0 otherwise. ⎩⎪

jωn

If we consider x[n ] = e as input to this system. Then we have the output

y[n ] = e jω ( n − nd ) = e − jωnd e jωn „

Therefore, the frequency response of the ideal delay is

„

Real and imaginary representation H ( e jω ) = cos(ωnd ) + j sin(ωnd ). Magnitude and phase representation

„

The frequency response is

H ( e jω ) =

H ( e jω ) = e − jωnd .

„

H ( e jω ) = 1 and ∠H ( e jω ) = −ωnd .

M2 1 e − jωn ∑ M 1 + M 2 + 1 n = − M1

=

1 e jωM1 − e − jω ( M 2 +1) M1 + M 2 + 1 1 − e − jω

=

1 sin(ω ( M 1 + M 2 + 1) / 2) − jω ( M 2 − M1 ) / 2 e . M1 + M 2 + 1 sin(ω / 2)

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Example 2.20 Frequency Response of the Moving-Average System (Cont’d)

Application Example - Signal Smoothing A common DSP application is the removal of noise from a signal corrupted by additive noise. A simple 3-point moving average algorithm is given by:

Symmetry Properties of the Fourier Transform

„ „

xo [n ] = − x [ − n ] Any sequence can be expressed as a sum of a conjugatesymmetric and conjugate-antisymmetric sequence * o

Conjugate-antisymmetric sequence

[ [

] ]

1 x[n ] + x * [ −n ] = x e* [ −n ] 2 1 xo [n ] = x[n ] − x * [ − n ] = − xo* [ − n ] 2 x e [n ] =

x[n ] = x e [n ] + x0 [n ]

„

Even sequence (real):

x e [n ] = x e [ − n ]

„

Odd sequence (real):

x o [ n ] = − xo [ − n ]

d[n] s[n] x[n]

Amplitude

6 4 2 0 -2

0

5

10

15

20 25 30 Time index n

35

40

45

50

8 y[n] s[n]

6 4 2 0

0

5

10

15

20 25 30 Time index n

35

40

45

50

Similarly, a Fourier transform X ( e jω ) can be decomposed into a sum of conjugate-symmetric and conjugate-antisymmetric functions. X ( e jω ) = X e ( e jω ) + X o ( e jω )

x e [n ] = x e* [ − n ]

Conjugate-symmetric sequence

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Symmetry Properties of the Fourier Transform (Cont’d)

Symmetry properties of the Fourier transform are often very useful for simplifying the solution of problems. Some basic definitions „

1 y[n] = ( x[n − 1] + x[n] + x[ n + 1]) 3

Amplitude

H ( e − jω0 ) = H * ( e jω0 )?

% Signal Smoothing by Averaging clf; % Generate random noise, d[n] R = 51; d = 0.8*(rand(R,1) - 0.5); % Generate uncorrupted signal, s[n] m = 0:R-1; s = 2*m.*(0.9.^m); % Generate noise corrupted signal, x[n] x = s + d'; subplot(2,1,1); plot(m,d','r-',m,s,'g--',m,x,'b-.'); xlabel('Time index n');ylabel('Amplitude'); legend('d[n] ','s[n] ','x[n] '); % do smoothing x1 = [0 0 x];x2 = [0 x 0];x3 = [x 0 0]; y = (x1 + x2 + x3)/3; subplot(2,1,2); plot(m,y(2:R+1),'r-',m,s,'g--'); legend( 'y[n] ','s[n] '); xlabel('Time index n');ylabel('Amplitude');

X e ( e jω ) =

[

]

1 X ( e jω ) + X * ( e − jω ) = X e* ( e − jω ) 2

Re{ x[n ]} =

1 ( x[n ] + x * [n ]) 2

X o ( e jω ) =

[

]

1 X ( e jω ) − X * ( e − jω ) = − X o* ( e − jω ) 2

j Im{x[n ]} =

1 ( x[n ] − x * [n ]) 2

For real sequences, the real part of the Fourier transform is an even function, and the imaginary part is an odd function.

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Fourier Transform Theorems

Frequency Response of LCCDE

Example: Determining an h[n] from Difference Equation y[n] −

H (e

H (e

H (e From

jω

jω

jω

1 1 y[n − 1] = x[n] − x[n − 1] 2 4 1 1 − jω jω H (e ) = 1 − e − )− e 4 2 1 1 − e − jω 4 ) = 1 1 − e − jω 2 1 − jω e 1 ) = − 4 1 − jω 1 − jω 1− e 1− e 2 2 Transform Tables n

⎛ 1 ⎞⎛ 1 ⎞ ⎛1⎞ h [ n ] = ⎜ ⎟ u [n ] − ⎜ ⎟ ⎜ ⎟ ⎝ 4 ⎠⎝ 2 ⎠ ⎝2⎠

n −1

jω

u [n − 1 ]

5

Example 2.22 Determining the Impulse Response from the Frequency Response The frequency response of a high-pass filter with delay

⎧e − jωnd H ( e jω ) = ⎨ ⎩ 0

ωc < ω < π . ω < ωc

H ( e jω ) = e − jωnd (1 − H lp ( e jω )) = e − jωnd − e − jωnd H lp ( e jω ), ⎧1, ω < ωc H lp ( e jω ) = ⎨ 0 , ω c < ω ≤ π. ⎩

h[n ] = δ [n − nd ] −

hlp [n ] =

sin ωc n πn

sin ωc ( n − nd ) π (n − nd )

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