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Course by: Thad Welch. E-mail the author

# DTFT and Convolution

Module by: Richard Baraniuk. E-mail the author

Summary: An overview of the DTFT and Convolution.

## Given:

Xeiω X ω and Heiω H ω , compute Yeiω Y ω
Yeiω=n =yne(i)ωn=n =k =xkhnke(iωn)=k =xkn =hnke(iωn) Y ω n y n ω n n k x k h n k ω n k x k n h n k ω n
(1)
Let (m=nk)(n=m+k) m n k n m k
=k =xkm =hme(iω(m+k))=k =xke(i)ωkm =hme(i)ωm k x k m h m ω m k k x k ω k m h m ω m
(2)
Yeiω=XeiωHeiω Y ω X ω H ω
(3)
x*h DTFT XH x h DTFT X H
(4)

## Example 1: Lowpass Filtering of an ECG Signal to Suppress Noise

ecg
• voltages are very small ( μV μ V !) therefore there is plenty of measurement noise
• use DSP to reduce noise
• simple processing: low pass filtering
Measurement process takes the TRUE SIGNAL and corrupts it with additive NOISE so that the actual received signal is SIGNAL+NOISE This is the signal we actually measure. It is "noisy".

### Key Point:

Signal is in low frequencies and noise is spread over all frequencies
We need a lowpass filter to suppress noise.

Try a simple averaging filter

### Averaging Filter h Physical Interpretation

h[n] h[n] is a moving average

• smooths the data
• h[n] h[n] is a LOWPASS FILTER
We see this behavior in Hω H ω . Also as 2M+1 2 M 1 increases, the cutoff frequency ω 1 =π2M+1 ω 1 2 M 1 decreases which leads to more smoothing. LOWPASS FILTERING = SMOOTHING = AVERAGING

### Result

Bottom Line: Figure 9 looks much better than Figure 6

#### At home:

1. Interpret the filtering action in terms of hn h n (time/sample domain) and Hω H ω (frequency domain)
2. What price did we pay?

## Extension 1:

Can we improve on the frequency response of the moving average filter?

## Extension 2:

ECG also picks up plenty of 60Hz interference

How would you combat this interference using an LTI filter?

## Glossary

electrocardiogram:
measure electrical signal given off by the heart.

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