Concept of combining multiple FIR Filters into 1 FIR filter

I try to combine my 7 band FIR filters into 1 FIR in order to process an audio signal. I use convolution to combine their coefficients but the effective is not the same as I apply 7 different FIR filter. I also have the gain problem

H = conv(conv(conv(conv(conv(conv(h1,h2),h3),h4),h5),h6),h7)

My questions are Is convolution a correct way to do combine multiple FIR filters into one filter If not, then what technique should I look into? If yes, then what did I do wrong?

Regards

Answer

What you tried to do in your example is combine the filters in series rather than parallel.
That is, if your signal is $$y = ((x*h_1)*h_2)\ldots*h_7$$ then, according to the associativity property of the Convolution, $$y=x*\underset{H}{\underbrace{(h_1*h_2\ldots*h_7)}}$$

However, if you wish to perform a parallel filtering then your signal is $$y=\sum_{i=1}^7 x*h_i$$ and acording to the distributivity property of the Convolution, $$y=x*\underset{H}{\underbrace{\sum_{i=1}^7 h_i}}$$