Sampling Rate Conversion by a Rational Factor I/D MCQ [Free PDF] – Objective Question Answer for Sampling Rate Conversion by a Rational Factor I/D Quiz

A sampling rate conversion by the rational factor I/D is accomplished by cascading an interpolator with a decimator.

We emphasize that the importance of performing the interpolation first and decimation second is to preserve the desired spectral characteristics of x(n).

In the diagram given, an interpolator is in a cascade with a decimator which together performs the action of sampling rate conversion by a factor I/D.

We know that the Discrete Fourier transform of a signal x(n) is given as

X(k)=\(\sum_{n=0}^{N-1} x(n)e^{-j2πkn/N}=\sum_{n=0}^{N-1} x(n) W_N^{kn}\)

Thus we get Nth rot of unity WN= e-j2π/N

The formula for calculating N point DFT is given as

X(k)=\(\sum_{n=0}^{N-1} x(n)e^{-j2πkn/N}\)

From the formula given at every step of computing, we are performing N complex multiplications and N-1 complex additions. So, in a total to perform N-point DFT we perform N2 complex multiplications and N(N-1) complex additions.

If XN represents the N point DFT of the sequence xN in the matrix form, then we know that X_N = W_N.x_N
By pre-multiplying both sides by W_N^-1, we get
x_N=W_N-1.X_N

But we know that the inverse DFT of X_N is defined as
x_N=1/N*X_N

Thus by comparing the above two equations we get
W_N-1=1/N W_N*

The first step is to determine the matrix W4. By exploiting the periodicity property of W4 and the symmetry property
\(W_{N}^{k+N/2}=-W_{N^k}\)

The matrix W4 may be expressed as

W4=\(\begin{bmatrix}W_4^0&W_4^0&W_4^0&W_4^1\\W_4^0&W_4^0&W_4^2&W_4^3\\W_4^0&W_4^2&W_4^0&W_4^3\\W_4^4&W_4^6&W_4^6&W_4^9\end{bmatrix}=\begin{bmatrix}W_4^0&W_4^0&W_4^0&W_4^1\\W_4^0&W_4^0&W_4^2&W_4^3\\W_4^0&W_4^2&W_4^0&W_4^3\\W_4^0&W_4^2&W_4^2&W_4^1\end{bmatrix}\)
=\(\begin{bmatrix}1&1&1&1\\1&-j&-1&j\\1&-1&1&-1\\1&j&-1&-j\end{bmatrix}\)

Then X4=W4.x4=\(\begin{bmatrix}6\\-2+2j\\-2\\-2-2j\end{bmatrix}\)

ck=\(\frac{1}{N} \sum_{n=0}^{N-1} x(n)e^{-j2πkn/N} = \frac{1}{N}X(k)=> X(k)=Nc_k\)

Given x(n)={0,1,2,3}

We know that the 4-point DFT of the above given sequence is given by the expression

X(k)=\(\sum_{n=0}^{N-1} x(n)e^{-j2πkn/N} \)

In this case N=4

=>X(0)=6, X(1)=-2+2j, X(2)=-2, X(3)=-2-2j.

We know that according to the periodicity and symmetry property,
100/4=200/x=>x=8.

Answer: A

As in the analog domain, frequency transformations can be performed on a digital low pass filter to convert it to either a bandpass, band stop, or high pass filter. The transformation involves the replacing of the variable z-1 with a rational function g(z-1).

Answer: C

The map z-1 → g(z-1) must map inside the unit circle in the z-plane into itself to apply digital frequency transformation.

Answer: B

For the map z-1 → g(z-1) to be a valid digital frequency transformation, then the unit circle also must be mapped inside the unit circle.

Answer: D

We know that the unit circle must be mapped inside the unit circle.
Thus it implies that for r=1, e-jω = g(e-jω)=|g(ω)|.ej arg [ g(ω) ]
It is clear that we must have |g(ω)|=1 for all ω. That is, the mapping is all-pass.

Answer: A

The value of |ak| < 1 to ensure that a stable filter is transformed into another stable filter to satisfy condition 1.

Answer: C

We know that the impulse invariance method and mapping of derivatives are inappropriate to use in the designing of high pass and bandpass filters.

Answer: A

We know that the impulse invariance method and mapping of derivatives are inappropriate to use in the designing of high pass and bandpass filters due to the aliasing problems.

Answer: B

Since there is a problem with aliasing in designing high pass and many bandpass filters using impulse invariance and mapping of derivatives, we cannot employ the analog frequency transformation followed by conversion of the result into the digital domain by use of these two mappings.

Answer: A

It is better to perform the mapping from an analog low pass filter into a digital low pass filter by either of these mappings and then perform the frequency transformation in the digital domain because, by this kind of frequency transformation, the problem of aliasing is avoided.

Answer: C

In the case of bilinear transformation, where aliasing is not a problem, it does not matter whether the frequency transformation is performed in the analog domain or in the frequency domain.

The design method based on the use of windows to truncate the impulse response h(n) and obtain the desired spectral shaping, was the first method proposed for designing linear phase FIR filters.

The major disadvantage of the window design method is the lack of precise control of the critical frequencies.

The values of the cutoff frequencies of a filter in general by the windowing technique depend on the type of the filter and the length of the filter.

In the frequency sampling technique, the transition band is a multiple of 2π/M.

The frequency sampling design method is particularly attractive when the FIR is realized either in the frequency domain by means of the DFT or in any of the frequency sampling realizations.

The attractive feature of the frequency sampling design is that the frequency response can take either zero or one at all frequencies, except in the transition band.

The Chebyshev approximation method provides total control of the filter specifications, and as a consequence, it is usually preferable to the other two methods.

By spreading the approximation error over the passband and stopband of the filter, this method results in an optimal filter design and using this the maximum sidelobe level is minimized.

The expression for Δf i.e., for the transition band is given as
Δf=(ωs-ωp)/2π.

The estimate is used to carry out the design and if the resulting δ exceeds the specified δ2, then the length can be increased until we obtain a side lobe level that meets the specification.