This topic deals the transforms, DFT and proof of its properties

1. Transforms
M.Starwin

2. Need for Transform
• Transforms make some types of calculations
much simpler and more convenient.
• Transforms are tools to make analysis easier.
• Laplace transform, for example, makes solving
differential equations easier.
• Possible to do all computation and analysis of
a signal in either the time domain or the
frequency domain.

3. Fourier Transform
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4. Discrete Time Fourier Transform Pair
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Synthesis Inverse Fourier Transform
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Fourier Transform
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5. Discrete Fourier Transform

6. DFT

7. IDFT

8. DFT PROPERTIES
Periodicity
If 𝑥 𝑛 ↔ 𝑋 𝑘
𝑥(𝑛 + 𝑁) = 𝑥 𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛
𝑋 𝑘 + 𝑁 = 𝑋 𝑘 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑘
DFT is defined as 𝑋 𝑘 =
𝑛=0
𝑁−1
𝑥(𝑛)𝑒−
𝑗2𝜋𝑛𝑘
𝑁
, 0≤𝑘≤𝑁−1
Sub k=k+N 𝑋 𝑘 + 𝑁 = 𝑛=0
𝑁−1
𝑥(𝑛)𝑒−
𝑗2𝜋𝑛(𝑘+𝑁)
𝑁

9. Cond..
𝑋 𝑘 + 𝑁 =
𝑛=0
𝑁−1
𝑥(𝑛)𝑒−
𝑗2𝜋𝑛𝑘
𝑁 𝑒−
𝑗2𝜋𝑛𝑁
𝑁
𝑋 𝑘 + 𝑁 =
𝑛=0
𝑁−1
𝑥(𝑛)𝑒−
𝑗2𝜋𝑛𝑘
𝑁
[𝑒−𝑗2𝜋𝑛
= 1]
𝑋 𝑘 + 𝑁 = 𝑋 𝑘
Hence Proved.

10. Linearity
• If 𝑥1 𝑛 ↔ 𝑋1 𝑘 𝑎𝑛𝑑 𝑥2 𝑛 ↔ 𝑋2 𝑘
Then ∝ 𝑥1 𝑛 + β𝑥2 𝑛 ↔ α𝑋1 𝑘 + β𝑋2 𝑘
DFT is defined as 𝑋 𝑘 = 𝑛=0
𝑁−1
𝑥(𝑛)𝑒−
𝑗2𝜋𝑛𝑘
𝑁
=
𝑛=0
𝑁−1
[∝ 𝑥1 𝑛 + β𝑥2 𝑛 ]𝑒−
𝑗2𝜋𝑛𝑘
𝑁
=
𝑛=0
𝑁−1
∝ 𝑥1 𝑛 𝑒−
𝑗2𝜋𝑛𝑘
𝑁 +
𝑛=0
𝑁−1
β𝑥2 𝑛 𝑒−
𝑗2𝜋𝑛𝑘
𝑁
=∝
𝑛=0
𝑁−1
𝑥1 𝑛 𝑒−
𝑗2𝜋𝑛𝑘
𝑁 + β
𝑛=0
𝑁−1
𝑥2 𝑛 𝑒−
𝑗2𝜋𝑛𝑘
𝑁
=α𝑋1 𝑘 + β𝑋2 𝑘
Hence proved

11. Circular Time Shifting
If 𝑥 𝑛 ↔ 𝑋 𝑘
then 𝑥(𝑛 − 𝑚) ↔= 𝑋 𝑘 𝑒−𝑗2𝜋𝑚𝑘
DFT is defined as 𝑋 𝑘 = 𝑛=0
𝑁−1
𝑥(𝑛)𝑒−
𝑗2𝜋𝑛𝑘
𝑁
=
𝑛=0
𝑁−1
𝑥(𝑛 − 𝑚)𝑒−
𝑗2𝜋𝑛𝑘
𝑁
Sub n-m=p, n=p+m
𝑋 𝑘 =
𝑝=0
𝑁−1
𝑥(𝑝)𝑒−
𝑗2𝜋(𝑝+𝑚)𝑘
𝑁
=
𝑝=0
𝑁−1
𝑥(𝑝)𝑒−
𝑗2𝜋𝑝𝑘
𝑁 𝑒−
𝑗2𝜋𝑚𝑘
𝑁
= 𝑒−
𝑗2𝜋𝑚𝑘
𝑁 𝑋 𝑘
Hence proved

12. Time Reversal property
If 𝑥 𝑛 ↔ 𝑋 𝑘
then 𝑥(−𝑛) ↔= 𝑋 −𝑘
DFT is defined as 𝑋 𝑘 = 𝑛=0
𝑁−1
𝑥(𝑛)𝑒−
𝑗2𝜋𝑛𝑘
𝑁
=
𝑛=0
𝑁−1
𝑥(−𝑛)𝑒−
𝑗2𝜋𝑛𝑘
𝑁
Sub –n=p
= 𝑝=0
𝑁−1
𝑥(𝑝)𝑒−
𝑗2𝜋(−𝑝)𝑘
𝑁
=
𝑛=0
𝑁−1
𝑥(−𝑛)𝑒−
𝑗2𝜋𝑝(−𝑘)
𝑁
= 𝑋(−𝑘)
Hence Proved

13. Circular Convolution Property
If 𝑥1(𝑛) ↔ 𝑋𝑘 and 𝑥2 𝑛 ↔ 𝑋2 𝑘
then 𝑥1 𝑛 ∗ 𝑥2 𝑛 ↔ 𝑋1 𝑘 . 𝑋2 𝑘
DFT is defined as 𝑋 𝑘 = 𝑛=0
𝑁−1
𝑥(𝑛)𝑒−
𝑗2𝜋𝑛𝑘
𝑁
=
𝑛=0
𝑁−1
𝑥1 𝑛 ∗ 𝑥2 𝑛 ]𝑒−
𝑗2𝜋𝑛𝑘
𝑁
=
𝑛=0
𝑁−1
𝑚=0
𝑁−1
𝑥1 𝑚 𝑥2(𝑛 − 𝑚) 𝑒−
𝑗2𝜋𝑛𝑘
𝑁
[𝑥1 𝑛 ∗ 𝑥2 𝑛 = 𝑛=0
𝑁−1
𝑥1 𝑚 𝑥2 𝑛 − 𝑚 ]

14. Cond..
=
𝑚=0
𝑁−1
𝑥1(𝑚)
𝑛=0
𝑁−1
𝑥2(𝑛 − 𝑚) 𝑒−𝑗2𝜋𝑛𝑘/𝑁
= 𝑋2(k) 𝑚=0
𝑁−1
𝑥1(𝑚) 𝑒−𝑗2𝜋𝑚𝑘/𝑁
[using
time shift]
= 𝑋2 𝐾 𝑋1(𝐾)
Hence proved

15. Multiplication property
If 𝑥1(𝑛) ↔ 𝑋1(𝐾) and 𝑥2 𝑛 ↔ 𝑋2 𝑘
then 𝑥1 𝑛 𝑥2 𝑛 ↔ 1/𝑁[𝑋1 𝑘 ∗ 𝑋2 𝑘 ]
DFT is defined as 𝑋 𝑘 = 𝑛=0
𝑁−1
𝑥(𝑛)𝑒−
𝑗2𝜋𝑛𝑘
𝑁
=
𝑛=0
𝑁−1
𝑥1 𝑛 𝑥2 𝑛 𝑒−
𝑗2𝜋𝑛𝑘
𝑁
We know 𝑥1 𝑛 = 1/𝑁 𝑙=0
𝑁−1
𝑋1 𝑙 𝑒
𝑗2𝜋𝑛𝑙
𝑁
𝑋 𝑘 =
𝑛=0
𝑁−1
1/𝑁
𝑙=0
𝑁−1
𝑋1 𝑙 𝑒
𝑗2𝜋𝑛𝑙
𝑁 𝑥2 𝑛 𝑒−
𝑗2𝜋𝑛𝑘
𝑁

16. Cond..
=
1
𝑁
𝑙=0
𝑁−1
𝑋1(𝑙)
𝑛=0
𝑁−1
𝑥2(𝑛)𝑒−𝑗2𝜋𝑛(𝑘−𝑙)/𝑁
=
1
𝑁
𝑙=0
𝑁−1
𝑋1(𝑙)𝑋2(𝑘 − 𝑙)
=
1
𝑁
[𝑋1 𝑙 ∗ 𝑋2 𝑙 ]
=
1
𝑁
[𝑋1 𝑘 ∗ 𝑋2 𝑘 ]
Hence Proved.