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.

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
dx —————-(1)
The inverse fourier transform of f(x) is defined by
f(x)=  ?F(u)  e
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

|F(u)| = R2
?(u) = tan-1
Spectral density is defined by
p(u) = |F(u)|

p(u) = R2

46. Give the relation for 1-D discrete fourier transform pair?
The discrete fourier transform is defined by
F(u) = 1/N  ?    f(x) e

The inverse discrete fourier transform is given by
f(x) =     ?     F(u) e
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.
F(u, v) =1/N  ? F(x, v) e
F(x, v)=N[1/N ?  f(x, y) e
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
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,
C(u)=?(u)? f(x) cos[((2x+1)u?)/2N]  where u=0,1,2,….N-1
Inverse f(x)= ? ?(u) c(u) cos[((2x+1) u?)/2N]  where x=0,1,2,…N-1

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