41.** What are the applications of transform. **

1) To reduce band width

2) To reduce redundancy

3) To extract feature.

42. **Give the Conditions for perfect transform? **

Transpose of matrix = Inverse of a matrix.

Orthoganality.

43. **What are the properties of unitary transform? **

1) Determinant and the Eigen values of a unitary matrix have unity magnitude 2) the entropy of a random vector is preserved under a unitary Transformation

3) Since the entropy is a measure of average information, this means information

is preserved under a unitary transformation.

44. **Define fourier transform pair? **

The fourier transform of f(x) denoted by F(u) is defined by

?

F(u)= ? f(x) e

-j2?ux

dx —————-(1)

-?

The inverse fourier transform of f(x) is defined by

?

f(x)= ?F(u) e

j2?ux

dx ——————–(2)

-?

The equations (1) and (2) are known as fourier transform pair.

45. **Define fourier spectrum and spectral density?**

Fourier spectrum is defined as

F(u) = |F(u)| e

j?(u)

Where

|F(u)| = R2

(u)+I

2

(u)

?(u) = tan-1

(I(u)/R(u))

Spectral density is defined by

p(u) = |F(u)|

2

p(u) = R2

(u)+I

2

(u)

46.** Give the relation for 1-D discrete fourier transform pair? **

The discrete fourier transform is defined by

n-1

F(u) = 1/N ? f(x) e

–j2?ux/N

x=0

The inverse discrete fourier transform is given by

n-1

f(x) = ? F(u) e

j2?ux/N

x=0

These equations are known as discrete fourier transform pair.

47. **Specify the properties of 2D fourier transform. **

The properties are

1. Separability

2. Translation

3. Periodicity and conjugate symmetry

4. Rotation

5. Distributivity and scaling

6. Average value

7. Laplacian

8. Convolution and correlation

9. sampling

48. **Explain separability property in 2D fourier transform **

The advantage of separable property is that F(u, v) and f(x, y) can be obtained by

successive application of 1D fourier transform or its inverse.

n-1

F(u, v) =1/N ? F(x, v) e

–j2?ux/N

x=0

Where

n-1

F(x, v)=N[1/N ? f(x, y) e

–j2?vy/N

y=0

49.** Properties of twiddle factor. **

1. Periodicity

WN^(K+N)= WN^K

2. Symmetry

WN^(K+N/2)= -WN^K

50. **Give the Properties of one-dimensional DFT **

1. The DFT and unitary DFT matrices are symmetric.

2. The extensions of the DFT and unitary DFT of a sequence and their

inverse transforms are periodic with period N.

3. The DFT or unitary DFT of a real sequence is conjugate symmetric

about N/2.

51. ** Give the Properties of two-dimensional DFT **

1. Symmetric

2. Periodic extensions

3. Sampled Fourier transform

4. Conjugate symmetry.

52. **What is meant by convolution? **

The convolution of 2 functions is defined by

f(x)*g(x) = f(?) .g(x- ?) d?

where ? is the dummy variable

53. **State convolution theorem for 1D **

If f(x) has a fourier transform F(u) and g(x) has a fourier transform G(u)

then f(x)*g(x) has a fourier transform F(u).G(u).

Convolution in x domain can be obtained by taking the inverse fourier

transform of the product F(u).G(u).

Convolution in frequency domain reduces the multiplication in the x

domain

F(x).g(x) F(u)* G(u)

These 2 results are referred to the convolution theorem.

54. **What is wrap around error? **

The individual periods of the convolution will overlap and referred to as

wrap around error

55.** Give the formula for correlation of 1D continuous function. **

The correlation of 2 continuous functions f(x) and g(x) is defined by

f(x) o g(x) = f*(? ) g(x+? ) d?

56. **What are the properties of Haar transform. **

1. Haar transform is real and orthogonal.

2. Haar transform is a very fast transform

3. Haar transform has very poor energy compaction for images

4. The basic vectors of Haar matrix sequensly ordered.

57. **What are the Properties of Slant transform
**

1. Slant transform is real and orthogonal.

2. Slant transform is a fast transform

3. Slant transform has very good energy compaction for images

4. The basic vectors of Slant matrix are not sequensely ordered.

58. **Specify the properties of forward transformation kernel? **

The forward transformation kernel is said to be separable if g(x, y, u, v)

g(x, y, u, v) = g1(x, u).g2(y, v)

The forward transformation kernel is symmetric if g1 is functionally equal to g2

g(x, y, u, v) = g1(x, u). g1(y,v)

59. **Define fast Walsh transform.
**

The Walsh transform is defined by

n-1 x-1

w(u) = 1/N ? f(x) ? (-1)

bi(x).bn-1-i (u)

x=0 i=0

60. **Give the relation for 1-D DCT. **

The 1-D DCT is,

N-1

C(u)=?(u)? f(x) cos[((2x+1)u?)/2N] where u=0,1,2,….N-1

X=0

N-1

Inverse f(x)= ? ?(u) c(u) cos[((2x+1) u?)/2N] where x=0,1,2,…N-1

V=0