Digital Signal Processing Quiz (2017-18)

The wrapping technique is a method used to design digital filters. It involves mapping the analog filter specifications onto a digital filter by wrapping the frequency response of the analog filter around the unit circle in the z-plane. This technique allows for the design of digital filters with desired characteristics, making the statement "we can Design Digital Filters using Wrapping Technique" true.

The DIT (Decimation in Time) algorithm is a method used in Fourier analysis to efficiently calculate the Discrete Fourier Transform (DFT). It divides the sequence into even and odd samples. This division helps in simplifying the calculation process by breaking down the sequence into smaller parts. By separating the even and odd samples, the algorithm can perform calculations on each subset individually and then combine the results to obtain the final DFT. This approach reduces the computational complexity and improves the efficiency of the algorithm.

A twiddle factor is a complex number used in the calculation of the discrete Fourier transform (DFT) and its inverse. It is used to rotate the input signal in the frequency domain. The twiddle factor is applied to each sample of the input signal, and its value depends on the frequency and position of the sample in the signal. By multiplying each sample with the appropriate twiddle factor, the DFT can be calculated efficiently. Therefore, WN is called the twiddle factor.

Linear convolution is a mathematical operation that combines two signals to produce a third signal. Circular convolution is a similar operation, but it assumes that the signals are periodic. In some cases, the linear convolution output can be obtained by applying the circular convolution method. This is because circular convolution can be used as an approximation for linear convolution when the signals are sufficiently long and the periodic assumption is not problematic. Therefore, the statement that linear convolution output can be obtained using/applying circular convolution method is true.

Periodicity refers to the recurring pattern or regular intervals in a sequence or set of elements. In this context, the property of periodicity supports the equation W88 = W80, indicating that there is a repeating pattern or cycle where the value of W88 is equal to the value of W80. This suggests that there is a predictable and consistent relationship between these two variables, occurring at regular intervals.

The DIT algorithm divides the sequence into even and odd samples. This means that it separates the elements of the sequence into two groups: one containing the even-indexed elements and the other containing the odd-indexed elements. This division allows for efficient computation of the Discrete Fourier Transform (DFT) by recursively splitting the sequence into smaller subproblems. By processing the even and odd samples separately, the DIT algorithm reduces the overall computational complexity and improves the efficiency of the Fourier transform calculation.

The statement is false because a 64-point FFT actually contains 6 stages of computations, not 5. In an FFT computation, the input sequence is divided into smaller sub-sequences, and each stage combines these sub-sequences in a specific way. In a 64-point FFT, the input sequence is divided into 2 sub-sequences of 32 points each in the first stage. Then, in each subsequent stage, the number of sub-sequences is doubled until the final stage, where the sub-sequences are combined to produce the final output. Therefore, there are 6 stages of computations in a 64-point FFT.

The circular convolution of two sequences in the time domain is equivalent to the multiplication of their Discrete Fourier Transforms (DFTs). This is a fundamental property of the Fourier Transform, where convolution in the time domain corresponds to multiplication in the frequency domain. By taking the DFT of the two sequences, multiplying them together, and then taking the inverse DFT, we obtain the circular convolution in the time domain.

The statement is false because the overlap save and overlap add methods do not require the same number of iteration blocks for computation of the final output. In the overlap save method, the input signal is divided into blocks, and each block is convolved with the filter. The output blocks are then overlapped and added to obtain the final output. In the overlap add method, the input signal is also divided into blocks, but the blocks are added with overlap, and then convolved with the filter. The number of iteration blocks required for computation can vary depending on the size of the input signal and the filter length.

The Discrete Fourier Transform (DFT) is preferred for its ability to determine the frequency component of the signal (1) and for filter design (3). The DFT allows us to analyze a signal in the frequency domain, which is useful for understanding the different frequency components present in the signal. Additionally, the DFT can be used to design filters, which are essential for removing unwanted frequency components or enhancing specific frequency ranges in a signal. Therefore, options 1 and 3 are correct.

Chebyshev and Butterworth filters do not give the same order for the same specifications in filter design. The order of a filter refers to the number of poles or zeros in the transfer function. Chebyshev filters have ripple in the passband and can achieve a steeper roll-off, which means they can have a lower order compared to Butterworth filters with the same specifications. Butterworth filters have a maximally flat response in the passband but have a slower roll-off, requiring a higher order to achieve the same level of attenuation as a Chebyshev filter. Therefore, the statement that Chebyshev and Butterworth filters give the same order for the same specifications is false.

The computational procedure for the Decimation in frequency algorithm takes Log2 N stages. This is because the algorithm involves dividing the input signal into smaller frequency bins, and each stage of the algorithm reduces the number of frequency bins by a factor of 2. Since the number of frequency bins is initially N, it takes Log2 N stages to reduce it to 1. Therefore, the correct answer is Log2 N stages.

The statement is false. Multiplication of two sequences in Fourier domain actually gives the circular convolution output in the time domain, not the linear convolution. Circular convolution is a periodic extension of the linear convolution, where the sequence wraps around at the boundaries. This distinction is important in applications such as signal processing and image filtering, where the choice of convolution type affects the results.

The Fast Fourier Transform (FFT) algorithm is commonly used to calculate the Discrete Fourier Transform (DFT) and the Inverse Discrete Fourier Transform (IDFT). Therefore, options 1) and 2) are correct. The Direct Z transform is a different mathematical transform and is not directly related to the FFT algorithm. Therefore, option 3) is incorrect. Option 4) states that all four options are correct, which is not true.

The correct answer is "Designing of digital filter in analog domain and transforming into digital domain". This process involves designing a digital filter using analog techniques and then converting it into the digital domain. This approach allows for the use of well-established analog filter design techniques while still obtaining the benefits of digital signal processing. The analog filter is designed to meet the desired specifications, and then it is transformed into the digital domain using methods such as discretization or bilinear transformation.

The overlap save method is a technique used to calculate the discrete convolution between a very long signal and a finite impulse response (FIR) filter. This method involves dividing the long signal into smaller segments, applying the FIR filter to each segment, and then overlapping and adding the filtered segments to obtain the final convolution result. By using the overlap save method, the computation can be performed efficiently and accurately for very long signals.

The transformations are required for analysis in time or frequency domain and for easier operations. These transformations allow for the conversion of signals between different representations, such as from the time domain to the frequency domain, which can provide valuable insights for analysis. Additionally, these transformations can simplify mathematical operations, making it easier to manipulate and process signals. Quantization and modulation are not directly related to the transformations mentioned in the question.

The output length of the linear convolution is determined by the sum of the number of samples in the first sequence and the number of samples in the second sequence, minus one. This is because during the linear convolution process, each sample in the first sequence is multiplied with all the samples in the second sequence, resulting in a total of M*N products. The output length should be able to accommodate all these products, hence the need to subtract one from the sum of the sample lengths.