6. Verify the orthogonality relation N-1 1 Σ e2nikn/Ne-2tiln/N N n=0 where Ski is the Kronecher's delta 1 {! 8kl = and Plancherel's identity = if k = 1 0 otherwise. Skl Hint: Write the exponential in terms of cosine and sine eix = cos(x) + i sin(x) when kl. Then use this orthogonality relation to prove Parceval's relation N-1 N-1 Σ f(n)g(n) = NΣ ƒ(k)ĝ(k) -NE n=0 k=0 N-1 N-1 Σ \f(n)|2 = N Σ \f(k)|2 n=0 k=0 for the discrete Fourier transform. Notice that Plancherel´s identity is a special case of Parceval's relation.
6. Verify the orthogonality relation N-1 1 Σ e2nikn/Ne-2tiln/N N n=0 where Ski is the Kronecher's delta 1 {! 8kl = and Plancherel's identity = if k = 1 0 otherwise. Skl Hint: Write the exponential in terms of cosine and sine eix = cos(x) + i sin(x) when kl. Then use this orthogonality relation to prove Parceval's relation N-1 N-1 Σ f(n)g(n) = NΣ ƒ(k)ĝ(k) -NE n=0 k=0 N-1 N-1 Σ \f(n)|2 = N Σ \f(k)|2 n=0 k=0 for the discrete Fourier transform. Notice that Plancherel´s identity is a special case of Parceval's relation.
Advanced Engineering Mathematics
10th Edition
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![### Orthogonality and Discrete Fourier Transform
This section will cover verifying orthogonality relations, and using this information to prove key identities in the discrete Fourier transform.
#### Verifying the Orthogonality Relation
The orthogonality relation is given by:
\[
\frac{1}{N} \sum_{n=0}^{N-1} e^{2\pi i kn/N} e^{-2\pi i ln/N} = \delta_{kl}
\]
where \( \delta_{kl} \) is the Kronecker's delta, defined as:
\[
\delta_{kl} =
\begin{cases}
1 & \text{if } k = l \\
0 & \text{otherwise}
\end{cases}
\]
**Hint:** Write the exponential in terms of cosine and sine: \( e^{ix} = \cos(x) + i \sin(x) \), especially when \( k \neq l \).
#### Using Orthogonality to Prove Key Identities
Use the verified orthogonality relation to derive the following identities:
1. **Parseval's Relation**:
\[
\sum_{n=0}^{N-1} f(n) g(n) = N \sum_{k=0}^{N-1} \hat{f}(k) \hat{g}(k)
\]
2. **Plancherel's Identity**:
\[
\sum_{n=0}^{N-1} |f(n)|^2 = N \sum_{k=0}^{N-1} |\hat{f}(k)|^2
\]
for the discrete Fourier transform.
**Note:** Plancherel's identity is a special case of Parseval's relation where \( f = g \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F979d0aba-5428-414f-a3ba-5510f0301082%2F5ee934a4-2026-4fd7-88ea-89bec6cfaffb%2Fv6kj5n_processed.png&w=3840&q=75)
Transcribed Image Text:### Orthogonality and Discrete Fourier Transform
This section will cover verifying orthogonality relations, and using this information to prove key identities in the discrete Fourier transform.
#### Verifying the Orthogonality Relation
The orthogonality relation is given by:
\[
\frac{1}{N} \sum_{n=0}^{N-1} e^{2\pi i kn/N} e^{-2\pi i ln/N} = \delta_{kl}
\]
where \( \delta_{kl} \) is the Kronecker's delta, defined as:
\[
\delta_{kl} =
\begin{cases}
1 & \text{if } k = l \\
0 & \text{otherwise}
\end{cases}
\]
**Hint:** Write the exponential in terms of cosine and sine: \( e^{ix} = \cos(x) + i \sin(x) \), especially when \( k \neq l \).
#### Using Orthogonality to Prove Key Identities
Use the verified orthogonality relation to derive the following identities:
1. **Parseval's Relation**:
\[
\sum_{n=0}^{N-1} f(n) g(n) = N \sum_{k=0}^{N-1} \hat{f}(k) \hat{g}(k)
\]
2. **Plancherel's Identity**:
\[
\sum_{n=0}^{N-1} |f(n)|^2 = N \sum_{k=0}^{N-1} |\hat{f}(k)|^2
\]
for the discrete Fourier transform.
**Note:** Plancherel's identity is a special case of Parseval's relation where \( f = g \).
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