Signal Processing Magic (6) -- Half Band Filter

Half-band filter is a FIR filter often used in decimation by 2 filtering operation. Benefit of this filter is that half of the filter coefficients are 0's, which reduces the complexity of filtering operation. The name of "half-band" comes from that the filter's cut-off frequency is roughly fs/4, thus the filter's pass band is around half of the total.

To design a half-band filter, it starts from a sinc function as \[s[n]=\frac{sin(\pi*n/2)}{\pi*n/2},n=-N,-(N-1),...,N\]
We know that when N approaches infinite, freq response of this sinc function is rectangle ranging from -fs/4 to fs/4. But in real world, N is always a limited number, and when that occurs, the filter does not look great. Ripples occur in pass band and rejection in stop band is often not good enough either.

How to improve this? Windowing method comes for rescue. As introduced before, windowing method enhances the pass band (main lobe) and suppresses the stop band (side lobes). Assuming window is w[n], after adding window, the new filter becomes
\[h[n]=s[n]*w[n]\]
Since half of \(s[n]\)'s coefficients are 0's, half of \(h[n]\)'s coefficients are 0's too.

Matlab script to generate a 21-tap half-band filter with Hamming window is shown below:

N = 10;
x = [-10 : 10]/2;
s = sinc(x)
w = hamming(2*N+1);
h = s.*w.';

figure;
subplot(2,1,1);stem(s);title('Sinc');
subplot(2,1,2);stem(h);title('Half-band');

[H, F] = freqz(s, 1, 'whole', 2^15);
[Hf, Ff] = freqz(h, 1, 'whole', 2^15);

figure; hold on;
plot(F(1:end/2)/pi/2, 20*log10(abs(H(1:end/2))));
plot(Ff(1:end/2)/pi/2, 20*log10(abs(Hf(1:end/2))));
grid;
ylabel('Magnitude (dB)'); xlabel('Freq (fs)');
legend('Sinc','Half-band');

Filter coefficients for Sinc and half-band filter

Freq response for Sinc and half-band filter

TensorFlow Android App Debugging (3) -- Keras to TensorFlow Model Conversion

Deep learning model can be programmed using different libraries. Among them, Keras is one of the most easy to use. However, network model generated using Keras can''t be directly applied to Android app. First it needs to be converted to TensorFlow. A conversion tool is provided here to automatically convert Keras model to TensorFlow. The instruction of this tool is straightforward. It converts .h5 (Keras model) to .pb (TensorFlow model).

    python keras_to_tensorflow.py 

        --input_model="path/to/keras/model.h5" 

        --output_model="path/to/save/model.pb"

For verification purpose, we can use tool named Tensorboard to visualize the generated model. As the first step, the model will be read into a folder by script read_pb_model.py. The default folder name is tf_summary.

python read_pb_model.py model.pb

Then tensorboard will convert the folder's content to a graph, and display it in a webpage such as http://nan-System-Product-Name:6006. Below is an instruction of tensorboard 

tensorboard --logdir ./tf_summary

An example of generated graph is shown below. The graph displays the network topology. It also shows the name of the node. It is important to know the names. In Android app, when one needs to access certain part of the network model, he has to use the name as reference.

For CV2 Color to Grayscale Image Conversion

The task is simple: we want to verify the formula used in CV2 (OpenCV 2.0) for converting color image to grayscale. The formula to convert from RGB to gray can be found here: Y = 0.299*R + 0.587*G + 0.114*B

We use the following code to verify the formula. First we read in a color image as both color (img_color) and grayscale (img_gray). Then we use the equation above to convert the color image to gray (img_color2gray). As the final step, we compare img_gray and img_color2gray. This comparison allows to verify the equation. The output of the equation is floating number. We use rounding (np.round) to convert float to integer. np.round(1.499) = 1 and np.round(1.5) = 2. We have tried other quantization methods such as floor or ceil but they don't behave as well.

img_name = 'lena_color_512.tif';
img_color = cv2.imread(img_name, cv2.IMREAD_COLOR)
img_gray = cv2.imread(img_name, cv2.IMREAD_GRAYSCALE)

shape = img_gray.shape
img_color2gray = np.zeros(shape, dtype=np.int32)
img_color2gray[:,:] = np.round(0.299*img_color[:,:,2]+0.587*img_color[:,:,1]+0.114*img_color[:,:,0])
img_color2gray = np.clip(img_color2gray, 0, 255)

The comparison result is this: img_gray and img_color2gray are very similar but not identical. Out of 512*512=262144 pixels in total, 352 pixels don't match (%0.13 of total). The dots in the figure below are the unmatched pixels. This leaves a question of how exactly this conversion is done in OpenCV SW. My source code can be found here.

Signal Processing Magic (5) -- Windowing

In discrete-time signal processing, if received signal is a tone, its spectrum looks like a sinc function (see this). The reason is by cutting digital signal to finite length, it is like imposing a rectangular window in time domain. And rectangular window in time domain corresponds to sinc function in freq domain.

But some engineers thought side lobes in sinc function are too big. In order to make them smaller, they invented windowing method, which is to impose a window on the time-domain signal and it can suppress side lobes in the frequency domain. Since then people have invented different types of windows (Bartlett, Hanning, Hamming, Blackman etc. see Sec 7.2 in Discrete-time signal processing written by Oppenheim&Schafer). Hamming and Hanning windows are shown below. Assuming the digital signal has M samples, to apply windowing, one should generate a window lasting M samples and then multiple window with digital sample by sample. Each window is well defined mathematically. For example, Hamming window for M samples is defined as:
\[w[n]=0.54-0.46cos(2\pi n/M)\]

Next we compare two signals, one with windowing and one without. The signal is a 100KHz tone with sampling rate of 1MHz. The plot below shows that by adding Hamming window, the side lobe is suppressed by around 20dB but main lobe becomes wider. Windowing function works like a magical rolling pin. It drives the signal energy from side lobes to the main lobe.

In the next plot, instead of one tone, we show the spectrum of two tones, one at 100KHz and another at 300KHz. After applying windowing, these two tones become more distinctive.