Question

Along with linear convolution, circular shift and circular convolution are fundamental to many digital signal processing algorithms. The circular shift by delay $n_0$ for a sequence of length N is denoted as: $x[(n - n_0)_N]$, where $(m)_N = m$ modulo N. Note: $(m)_N \in \{0, 1, \dots, N - 1\}$ for any m. Circular convolution is defined as $$x[n] * _N h[n] = \sum_{k=0}^{N-1} x[k]h[(n - k)_N],$$ where the N in $_N$ denotes the length of the sequence. Consider a signal and impulse response for a sequence of length N = 4: $x[n] = -2\delta[n] + \delta[n - 1] + 3\delta[n - 2]$, $h[n] = \delta[n] + 3\delta[n - 1]$. a) Write down the the sequence x[n] after a circular shift by 1 delay. b) Compute the linear convolution $x[n] h[n]$. c) Compute the circular convolution $x[n] * _3 h[n]$. d) Describe the relationship between circular convolution and the Discrete Fourier Transform (DFT).

          Along with linear convolution, circular shift and circular convolution are fundamental to many digital signal
processing algorithms. The circular shift by delay $n_0$ for a sequence of length N is denoted as:
$x[(n - n_0)_N]$, where $(m)_N = m$ modulo N.
Note: $(m)_N \in \{0, 1, \dots, N - 1\}$ for any m.
Circular convolution is defined as
$$x[n] * _N h[n] = \sum_{k=0}^{N-1} x[k]h[(n - k)_N],$$
where the N in $*_N$ denotes the length of the sequence.
Consider a signal and impulse response for a sequence of length N = 4:
$x[n] = -2\delta[n] + \delta[n - 1] + 3\delta[n - 2]$,
$h[n] = \delta[n] + 3\delta[n - 1]$.
a) Write down the the sequence x[n] after a circular shift by 1 delay.
b) Compute the linear convolution $x[n] * h[n]$.
c) Compute the circular convolution $x[n] * _3 h[n]$.
d) Describe the relationship between circular convolution and the Discrete Fourier Transform (DFT).

$Along with linear convolution, circular shift and circular convolution are fundamental to many digital signal processing algorithms. The circular shift by delay n0 for a sequence of length N is denoted as: x[(n - n0)N], where (m)N = m modulo N. Note: (m)N ∈{0, 1, …, N - 1} for any m. Circular convolution is defined as x[n] * N h[n] = ∑k=0^N-1 x[k]h[(n - k)N], where the N in *N denotes the length of the sequence. Consider a signal and impulse response for a sequence of length N = 4: x[n] = -2δ[n] + δ[n - 1] + 3δ[n - 2], h[n] = δ[n] + 3δ[n - 1]. a) Write down the the sequence x[n] after a circular shift by 1 delay. b) Compute the linear convolution x[n] * h[n]. c) Compute the circular convolution x[n] * 3 h[n]. d) Describe the relationship between circular convolution and the Discrete Fourier Transform (DFT).$

Added by Vincent C.

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