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DTFT and Convolution

Module by: Richard Baraniuk. E-mail the author

Summary: An overview of the DTFT and Convolution.

Figure 1
Figure 1 (fig1.png)

Given:

X⁢ei⁢ω X ω and H⁢ei⁢ω H ω , compute Y⁢ei⁢ω Y ω
Y⁢ei⁢ω=∑n =−∞∞y⁢n⁢e(−i)⁢ω⁢n=∑n =−∞∞∑k =−∞∞x⁢k⁢h⁢n−k⁢e−(i⁢ω⁢n)=∑k =−∞∞x⁢k⁢∑n =−∞∞h⁢n−k⁢e−(i⁢ω⁢n) 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=n−k)→(n=m+k) m n k n m k
=∑k =−∞∞x⁢k⁢∑m =−∞∞h⁢m⁢e−(i⁢ω⁢(m+k))=∑k =−∞∞x⁢k⁢e(−i)⁢ω⁢k⁢∑m =−∞∞h⁢m⁢e(−i)⁢ω⁢m k x k m h m ω m k k x k ω k m h m ω m
(2)
Y⁢ei⁢ω=X⁢ei⁢ω⁢H⁢ei⁢ω Y ω X ω H ω
(3)
Figure 2
Figure 2 (fig2.png)
x*h ↔ DTFT X⁢H x h ↔ DTFT X H
(4)

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

Figure 3
Figure 3 (fig3.png)
ecg
  • voltages are very small ( μ⁢V 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
Figure 4
Figure 4 (fig4.png)
and corrupts it with additive NOISE
Figure 5
Figure 5 (fig5.png)
so that the actual received signal is SIGNAL+NOISE
Figure 6
Figure 6 (fig6.png)
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

Figure 7
Figure 7 (fig7.png)

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 H⁢ω H ω . Also as 2⁢M+1 2 M 1 increases, the cutoff frequency ω 1 =π2⁢M+1 ω 1 2 M 1 decreases which leads to more smoothing.
Figure 8
Figure 8 (fig8.png)
LOWPASS FILTERING = SMOOTHING = AVERAGING

Result

Figure 9
Figure 9 (fig9.png)
Bottom Line: Figure 9 looks much better than Figure 6

At home:

  1. Interpret the filtering action in terms of h⁢n h n (time/sample domain) and H⁢ω H ω (frequency domain)
  2. What price did we pay?
Figure 10
SUMMARY
SUMMARY (fig10.png)

Extension 1:

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

Extension 2:

ECG also picks up plenty of 60Hz interference

Figure 11
Figure 11 (fig12.png)
How would you combat this interference using an LTI filter?

Glossary

electrocardiogram:
measure electrical signal given off by the heart.

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